0412664v1Quantum Transport in Semiconductor Nanostructures

8/10/2019 0412664v1Quantum Transport in Semiconductor Nanostructures

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phase coherence characteristic of a microscopic objectcan be maintained at low temperatures (below 1 K) overdistances of several microns, which one would otherwisehave classified as macroscopic. The physics of these sys-tems has been referred to as mesoscopic,1 a word bor-rowed from statistical mechanics.2 Elastic impurity scat-tering does not destroy phase coherence, which is whythe effects of quantum interference can modify the con-

ductivity of a disordered conductor. This is the regimeof diffusive transport, characteristic for disordered met-als. Quantum interference becomes more important asthe dimensionality of the conductor is reduced. Quasi-one dimensionality can readily be achieved in a 2DEG bylateral confinement.

Semiconductor nanostructures are unique in offeringthe possibility of studying quantum transport in an arti-ficial potential landscape. This is the regime ofballistictransport, in which scattering with impurities can be ne-glected. The transport properties can then be tailored byvarying the geometry of the conductor, in much the sameway as one would tailor the transmission properties of a

waveguide. The physics of this transport regime could becalledelectron opticsin the solid state.3 The formal rela-tion between conduction and transmission, known as theLandauer formula,1,4,5 has demonstrated its real powerin this context. For example, the quantization of the con-ductance of a quantum point contact6,7 (a short and nar-row constriction in the 2DEG) can be understood usingthe Landauer formula as resulting from the discretenessof the number of propagating modes in a waveguide.

Two-dimensional systems in a perpendicular magneticfield have the remarkable property of a quantized Hallresistance,8 which results from the quantization of theenergy in a series of Landau levels. The magnetic length(h/eB)1/2 (

10 nm at B = 5 T) assumes the role of the

wavelength in the quantum Hall effect. The potentiallandscape in a 2DEG can be adjusted to be smooth onthe scale of the magnetic length, so that inter-Landaulevel scattering is suppressed. One then enters the regimeof adiabatic transport. In this regime truly macroscopicbehavior may not be found even in samples as large as0.25 mm.

In this review we present a self-contained account ofthese three novel transport regimes in semiconductornanostructures. The experimental and theoretical de-velopments in this field have developed hand in hand, afruitful balance that we have tried to maintain here aswell. We have opted for the simplest possible theoreticalexplanations, avoiding the powerful but more formal Greens function techniques. If in some instances thischoice has not enabled us to do full justice to a subject,then we hope that this disadvantage is compensated bya gain in accessibility. Lack of space and time has causedus to limit the scope of this review to metallic transportin the plane of a 2DEG at small currents and voltages.Transport in the regime of strong localization is excluded,as well as that in the regime of a nonlinear current-voltage dependence. Overviews of these, and other, top-

ics not covered here may be found in Refs.9,10,11, as wellas in recent conference proceedings.12,13,14,15,16,17

We have attempted to give a comprehensive list ofreferences to theoretical and experimental work on thesubjects of this review. We apologize to those whosecontributions we have overlooked. Certain experimentsare discussed in some detail. In selecting these experi-ments, our aim has been to choose those that illustrate

a particular phenomenon in the clearest fashion, not toestablish priorities. We thank the authors and publishersfor their kind permission to reproduce figures from theoriginal publications. Much of the work reviewed herewas a joint effort with colleagues at the Delft Universityof Technology and at the Philips Research Laboratories,and we are grateful for the stimulating collaboration.

The study of quantum transport in semiconductornanostructures is motivated by more than scientific inter-est. The fabrication of nanostructures relies on sophis-ticated crystal growth and lithographic techniques thatexist because of the industrial effort toward the minia-turization of transistors. Conventional transistors oper-

ate in the regime of classical diffusive transport, whichbreaks down on short length scales. The discovery ofnovel transport regimes in semiconductor nanostructuresprovides options for the development of innovative fu-ture devices. At this point, most of the proposals inthe literature for a quantum interference device havebeen presented primarily as interesting possibilities, andthey have not yet been critically analyzed. A quanti-tative comparison with conventional transistors will beneeded, taking circuit design and technological consider-ations into account.18 Some proposals are very ambitious,in that they do not only consider a different principle ofoperation for a single transistor, but envision entire com-puter architectures in which arrays of quantum devices

operate phase coherently.19

We hope that the present review will convey some ofthe excitement that the workers in this rewarding field ofresearch have experienced in its exploration. May the de-scription of the variety of phenomena known at present,and of the simplest way in which they can be understood,form an inspiration for future investigations.

B. Nanostructures in Si inversion layers

Electronic properties of the two-dimensional electrongas in Si MOSFETs (metal-oxide-semiconductor field-effect transistors) have been reviewed by Ando, Fowler,and Stern,20 while general technological and device as-pects are covered in detail in the books by Sze21 and byNicollian and Brew.22 In this section we only summarizethose properties that are needed in the following. A typ-ical device consists of a p-type Si substrate, covered by aSi02 layer that serves as an insulator between the (100)Si surface and a metallic gate electrode. By applicationof a sufficiently strong positive voltage Vg on the gate, a2DEG is induced electrostatically in the p-type Si under


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FIG. 1 Band-bending diagram (showing conduction bandEc, valence band Ev, and Fermi level EF) of a metal-oxide-semiconductor (MOS) structure. A 2DEG is formed at theinterface between the oxide and the p-type silicon substrate,as a consequence of the positive voltage Von the metal gateelectrode.

the gate. The band bending leading to the formation ofthis inversion layer is schematically indicated in Fig. 1.The areal electron concentration (or sheet density)nsfol-lows from ens = Cox(Vg Vt), whereVt is the thresholdvoltage beyond which the inversion layer is created, andCoxis the capacitance per unit area of the gate electrodewith respect to the electron gas. Approximately, one hasCox = ox/dox (with ox = 3.9 0 the dielectric constantof the Si02 layer),

21 so

ns =

ox

edox (Vg Vt). (1.1)The linear dependence of the sheet density on the appliedgate voltage is one of the most useful properties of Siinversion layers.

The electric field across the oxide layer resulting fromthe applied gate voltage can be quite strong. Typically,Vg Vt = 5V and dox = 50 nm, so the field strengthis of order 1 MV/cm, at best a factor of 10 lower thantypical fields for the dielectric breakdown of Si02. It ispossible to change the electric field at the interface, with-out alteringns, by applying an additional voltage acrossthep-njunction that isolates the inversion layer from the

p-type substrate (such a voltage is referred to as a sub-strate bias). At the Si-Si02 interface the electric field iscontinuous, but there is an electrostatic potential step ofabout 3 eV. An approximately triangular potential wellis thus formed at the interface (see Fig. 1). The actualshape of the potential deviates somewhat from the tri-angular one due to the electronic charge in the inversionlayer, and has to be calculated self-consistently.20 Dueto the confinement in one direction in this potential well,the three-dimensional conduction band splits into a seriesof two-dimensional subbands. Under typical conditions

(for a sheet electron densityns = 1011 1012 cm2) only

a single two-dimensional subband is occupied. Bulk Sihas an indirect band gap, with six equivalent conductionband valleys in the (100) direction in reciprocal space. Ininversion layers on the (100) Si surface, the degeneracybetween these valleys is partially lifted. A twofold val-ley degeneracy remains. In the following, we treat thesetwo valleys as completely independent, ignoring compli-

cations due to intervalley scattering. For each valley, the(one-dimensional) Fermi surface is simply a circle, corre-sponding to free motion in a plane with effective electronmass20 m = 0.19 me. For easy reference, this and otherrelevant numbers are listed in Table I.

The electronic properties of the Si inversion layercan be studied by capacitive or spectroscopic techniques(which are outside the scope of this review), as well asby transport measurements in the plane of the 2DEG.To determine the intrinsic transport properties of the2DEG (e.g., the electron mobility), one defines a widechannel by fabricating a gate electrode with the appro-priate shape. Ohmic contacts to the channel are then

made by ion implantation, followed by a lateral diffusionand annealing process. The two current-carrying con-tacts are referred to as the source and the drain. One ofthese also serves as zero reference for the gate voltage.Additional side contacts to the channel are often fabri-cated as well (for example, in the Hall bar geometry), toserve as voltage probes for measurements of the longi-tudinal and Hall resistance. Insulation is automaticallyprovided by the p-n junctions surrounding the inversionlayer. (Moreover, at the low temperatures of interesthere, the substrate conduction vanishes anyway due tocarrier freeze-out.) The electron mobility e is an im-portant figure of merit for the quality of the device. Atlow temperatures the mobility in a given sample variesnonmonotonically20 with increasing electron density ns(or increasing gate voltage), due to the opposite effectsof enhanced screening (which reduces ionized impurityscattering) and enhanced confinement (which leads to anincrease in surface roughness scattering at the Si-Si02interface). The maximum low-temperature mobility ofelectrons in high-quality samples is around 104 cm2/Vs.This review deals with the modifications of the trans-port properties of the 2DEG in narrow geometries. Sev-eral lateral confinement schemes have been tried in orderto achieve narrow inversion layer channels (see Fig. 2).Many more have been proposed, but here we discuss onlythose realized experimentally.

Technically simplest, because it does not require elec-tron beam lithography, is an approach first used byFowler et al., following a suggestion by Pepper32,33,34

(Fig. 2a). By adjusting the negative voltage over p-njunctions on either side of a relatively wide gate, theywere able to vary the electron channel width as well asits electron density. This technique has been used to de-fine narrow accumulation layers onn-type Si substrates,rather than inversion layers. Specifically, it has been usedfor the exploration of quantum transport in the strongly


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TABLE I Electronic properties of the 2DEG in GaAs-AlGaAs heterostructures and Si inversion layers.

GaAs(100) Si(100) Units

Effective Mass m 0.067 0.19 me= 9.1 1028 gSpin Degeneracy gs 2 2

Valley Degeneracy gv 1 2

Dielectric Constant 13.1 11.9 0= 8.9 1012 Fm1

Density of States (E) = gsgv(m/2h

2

) 0.28 1.59 10

11

cm

2

meV

1

Electronic Sheet Densitya ns 4 110 1011 cm2

Fermi Wave Vector kF = (4ns/gsgv)1/2 1.58 0.561.77 106 cm1

Fermi Velocity vF = hkF/m 2.7 0.341.1 107 cm/s

Fermi Energy EF = (hkF)2/2m 14 0.636.3 meV

Electron Mobilitya e 104 106 104 cm2/Vs

Scattering Time =me/e 0.3838 1.1 ps

Diffusion Constant D= v2F /2 14014000 6.464 cm2/s

Resistivity = (nsee)1 1.60.016 6.30.63 k

Fermi Wavelength F = 2/kF 40 11235 nm

Mean Free Path l= vF 102 104 37118 nm

Phase Coherence Lengthb l= (D)1/2 200... 40400 nm(T /K)1/2

Thermal Length lT = (hD/kBT)1/2

3303300 70220 nm(T /K)1/2

Cyclotron Radius lcycl= hkF/eB 100 37116 nm(B/T)1

Magnetic Length lm = (h/eB)1/2 26 26 nm(B/T)1/2

kFl 15.81580 2.121

c 1100 1 (B/T)

EF/hc 7.9 110 (B/T)1

aA typical (fixed) density value is taken for GaAs-AlGaAs het-erostructures, and a typical range of values in the metallic con-duction regime for Si MOSFETs. For the mobility, a range ofrepresentative values is listed for GaAs-AlGaAs heterostructures,and a typical good value for Si MOSFETs. The variation in theother quantities reflects that inns and e.bRough estimate of the phase coherence length, based

on weak localization experiments in laterally confined

heterostructures23,24,25,26,27 and Si MOSFETs.28,29 The statedT1/2 temperature dependence should be regarded as an indica-tion only, since a simple power law dependence is not always found(see, for example, Refs.30 and25). For high-mobility GaAs-AlGaAsheterostructures the phase coherence length is not known, but ispresumably31 comparable to the (elastic) mean free path l .

localized regime32,35,36,37 (which is not discussed in thisreview). Perhaps the technique is particularly suited tothis highly resistive regime, since a tail of the diffusionprofile inevitably extends into the channel, providing ad-ditional scattering centers.34 Some studies in the weaklocalization regime have also been reported.33

The conceptually simplest approach (Fig.2b) to definea narrow channel is to scale down the width of the gate bymeans of electron beam lithography38 or other advancedtechniques.39,40,41 A difficulty for the characterization ofthe device is that fringing fields beyond the gate inducea considerable uncertainty in the channel width, as wellas its density. Such a problem is shared to some degreeby all approaches, however, and this technique has beenquite successful (as we will discuss in Section II). Fora theoretical study of the electrostatic confining poten-

tial induced by the narrow gate, we refer to the workby Laux and Stern.42 This is a complicated problem,which requires a self-consistent solution of the Poissonand Schrodinger equations, and must be solved numeri-cally.

The narrow gate technique has been modified by War-ren et al.43,44 (Fig.2c), who covered a multiple narrow-gate structure with a second dielectric followed by a sec-ond gate covering the entire device. (This structure wasspecifically intended to study one-dimensional superlat-tice effects, which is why multiple narrow gates wereused.) By separately varying the voltages on the twogates, one achieves an increased control over channelwidth and density. The electrostatics of this particu-lar structure has been studied in Ref.43 in a semiclassicalapproximation.


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FIG. 2 Schematic cross-sectional views of the lateral pinch-off technique used to define a narrow electron accumulationlayer (a), and of three different methods to define a narrowinversion layer in Si MOSFETs (b-d). Positive (+) and neg-ative () charges on the gate electrodes are indicated. Thelocation of the 2DEG is shown in black.

Skocpol et al.29,45 have combined a narrow gate witha deep self-aligned mesa structure (Fig. 2d), fabricatedusing dry-etching techniques. One advantage of theirmethod is that at least an upper bound on the channelwidth is known unequivocally. A disadvantage is thatthe deep etch exposes the sidewalls of the electron gas,so that it is likely that some mobility reduction occursdue to sidewall scattering. In addition, the deep etchmay damage the 2DEG itself. This approach has beenused successfully in the exploration of nonlocal quantumtransport in multiprobe channels, which in addition tobeing narrow have a very small separation of the voltage

probes.45,46 In another investigation these narrow chan-nels have been used as instruments sensitive to the charg-ing and discharging of a single electron trap, allowing adetailed study of the statistics of trap kinetics.46,47,48

C. Nanostructures in GaAs-AlGaAs heterostructures

In a modulation-doped49 GaAs-AlGaAs heterostruc-ture, the 2DEG is present at the interface between GaAsand AlxGa1xAs layers (for a recent review, see Ref.

50).Typically, the Al mole fraction x = 0.3. As shown in theband-bending diagram of Fig. 3, the electrons are con-fined to the GaAs-AlGaAs interface by a potential well,formed by the repulsive barrier due to the conductionband offset of about 0.3 V between the two semiconduc-tors, and by the attractive electrostatic potential due tothe positively charged ionized donors in the n-doped Al-GaAs layer. To reduce scattering from these donors, thedoped layer is separated from the interface by an un-doped AlGaAs spacer layer. Two-dimensional sub bandsare formed as a result of confinement perpendicular to theinterface and free motion along the interface. An impor-

FIG. 3 Band-bending diagram of a modulation doped GaAs-AlxGa1xAs heterostructure. A 2DEG is formed in the un-doped GaAs at the interface with the p-type doped AlGaAs.Note the Schottky barrier between the semiconductor and ametal electrode.

tant advantage over a MOSFET is that the present inter-face does not interrupt the crystalline periodicity. Thisis possible because GaAs and AlGaAs have almost thesame lattice spacing. Because of the absence of bound-ary scattering at the interface, the electron mobility canbe higher by many orders of magnitude (see Table I).The mobility is also high because of the low effectivemass m= 0.067 me in GaAs (for a review of GaAs ma-terial properties, see Ref.51). As in a Si inversion layer,only a single two-dimensional subband (associated withthe lowest discrete confinement level in the well) is usu-ally populated. Since GaAs has a direct band gap, witha single conduction band minimum, complications dueto intervalley scattering (as in Si) are absent. The one-dimensional Fermi surface is a circle, for the commonlyused (100) substrate orientation.

Since the 2DEG is present naturally due to the mod-ulation doping (i.e., even in the absence of a gate), thecreation of a narrow channel now requires the selectivedepletion of the electron gas in spatially separated re-gions. In principle, one could imagine using a combina-tion of an undoped heterostructure and a narrow gate(similarly to a MOSFET), but in practice this does notwork very well due to the lack of a natural oxide to serve

as an insulator on top of the AlGaAs. The Schottky bar-rier between a metal and (Al)GaAs (see Fig. 3) is toolow (only 0.9 V) to sustain a large positive voltage onthe gate. For depletion-type devices, where a negativevoltage is applied on the gate, the Schottky barrier isquite sufficient as a gate insulator (see, e.g., Ref.52).

The simplest lateral confinement technique is illus-trated in Fig.4a. The appropriate device geometry (suchas a Hall bar) is realized by defining a deep mesa, bymeans of wet chemical etching. Wide Hall bars are usu-


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FIG. 4 Schematic cross-sectional views of four different waysto define narrow 2DEG channels in a GaAs-AlGaAs het-erostructure. Positive ionized donors and negative charges ona Schottky gate electrode are indicated. The hatched squaresin d represent unremoved resist used as a gate dielectric.

ally fabricated in this way. This approach has also beenused to fabricate the first micron-scale devices, such asthe constrictions used in the study of the breakdown ofthe quantum Hall effect by Kirtley et al.53 and Bliek etal.,54 and the narrow channels used in the first study ofquasi-one-dimensional quantum transport in heterostruc-tures by Choi et al.55 The deep-mesa confinement tech-nique using wet25,56 or dry57 etching is still of use forsome experimental studies, but it is generally felt to beunreliable for channels less than 1m wide (in particularbecause of the exposed sidewalls of the structure).

The first working alternative confinement scheme wasdeveloped by Thornton et al.58 and Zheng et al.,24 whointroduced the split-gate lateral confinement technique(Fig.4b). On application of a negative voltage to a splitSchottky gate, wide 2DEG regions under the gate are de-pleted, leaving a narrow channel undepleted. The mostappealing feature of this confinement scheme is that thechannel width and electron density can be varied contin-uously (but not independently) by increasing the nega-tive gate voltage beyond the depletion threshold in thewide regions (typically about0.6 V). The split-gatetechnique has become very popular, especially after itwas used to fabricate the short and narrow constrictions

known as quantum point contacts6,7,59 (see SectionIII).The electrostatic confinement problem for the split-gategeometry has been studied numerically in Refs.60 and61.A simple analytical treatment is given in in Ref.62. Amodification of the split-gate technique is the grating-gate technique, which may be used to define a 2DEGwith a periodic density modulation.62

The second widely used approach is the shallow-mesadepletion technique (Fig.4c), introduced in Ref.63. Thistechnique relies on the fact that a 2DEG can be de-

pleted by removal of only a thin layer of the AlGaAs,the required thickness being a sensitive function of theparameters of the heterostructure material, and of de-tails of the lithographic process (which usually involveselectron beam lithography followed by dry etching). Theshallow-mesa etch technique has been perfected by twogroups,64,65,66 for the fabrication of multi probe electronwaveguides and rings.67,68,69,70 Submicron trenches71 are

still another way to define the channel. For simple analyt-ical estimates of lateral depletion widths in the shallow-mesa geometry, see Ref.72.

A clever variant of the split-gate technique was intro-duced by Ford et al.73,74 A patterned layer of electronbeam resist (an organic insulator) is used as a gate di-electric, in such a way that the separation between thegate and the 2DEG is largest in those regions where a nar-row conducting channel has to remain after applicationof a negative gate voltage. As illustrated by the cross-sectional view in Fig.4d, in this way one can define a ringstructure, for example, for use in an Aharonov-Bohm ex-periment. A similar approach was developed by Smith

et al.75

Instead of an organic resist they use a shallow-mesa pattern in the heterostructure as a gate dielectricof variable thickness. Initially, the latter technique wasused for capacitive studies of one- and zero-dimensionalconfinement.75,76 More recently it was adopted for trans-port measurements as well.77 Still another variation ofthis approach was developed by Hansen et al.,78,79 pri-marily for the study of one-dimensional subband struc-ture using infrared spectroscopy. Instead of electronbeam lithography, they employ a photolithographic tech-nique to define a pattern in the insulator. An array witha very large number of narrow lines is obtained by pro-

jecting the interference pattern of two laser beams ontolight-sensitive resist. This technique is known as holo-graphic illumination (see SectionII.G.2).

As two representative examples of state-of-the-artnanostructures, we show in Fig. 5a a miniaturized Hallbar,67 fabricated by a shallow-mesa etch, and in Fig. 5ba double-quantum-point contact device,80 fabricated bymeans of the split-gate technique.

Other techniques have been used as well to fabricatenarrow electron gas channels. We mention selective-areaion implantation using focused ion beams,81 masked ionbeam exposure,82 strain-induced confinement,83 lateral

p-n junctions,84,85 gates in the plane of the 2DEG,86

and selective epitaxial growth.87,88,89,90,91,92 For more de-tailed and complete accounts of nanostructure fabrica-tion techniques, we refer to Refs.9 and13,14,15.

D. Basic properties

1. Density of states in two, one, and zero dimensions

The energy of conduction electrons in a single subbandof an unbounded 2DEG, relative to the bottom of that


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FIG. 5 Scanning electron micrographs of nanostructuresin GaAs-AlGaAs heterostructures. (a, top) Narrow chan-nel (width 75 nm), fabricated by means of the confinementscheme of Fig. 4c. The channel has side branches (at a 2-mseparation) that serve as voltage probes. Taken from M. L.Roukes et al., Phys. Rev. Lett. 59, 3011 (1987). (b, bottom)Double quantum point contact device, based on the confine-ment scheme of Fig. 4b. The bar denotes a length of 1 m.Taken from H. van Houten et al., Phys. Rev. B 39, 8556(1989).

subband, is given by

E(k) = h2k2/2m, (1.2)

as a function of momentum hk. The effective mass mis considerably smaller than the free electron mass me(see TableI), as a result of interactions with the latticepotential. (The incorporation of this potential into aneffective mass is an approximation20 that is completely

justified for the present purposes.) The density of states(E) dn(E)/dEis the derivative of the number of elec-tronic states n(E) (per unit surface area) with energysmaller than E. In k-space, these states are containedwithin a circle of area A = 2mE/h2 [according to Eq.(1.2)], which contains a number gsgvA/(2)

2 of distinctstates. The factorsgs andgv account for the spin degen-eracy and valley degeneracy, respectively (Table I). Onethus finds that n(E) = gsgvmE/2h

2, so the density of

states corresponding to a single subband in a 2DEG,

(E) = gsgvmE/2h2, (1.3)

is independent of the energy. As illustrated in Fig. 6a,a sequence of subbands is associated with the set of dis-crete levels in the potential well that confines the 2DEGto the interface. At zero temperature, all states are

filled up to the Fermi energy EF (this remains a goodapproximation at finite temperature if the thermal en-ergy kBT EF). Because of the constant density ofstates, the electron (sheet) density ns is linearly relatedto EF by ns = EFgsgvm/2h

2. The Fermi wave num-ber kF = (2mEF/h

2)1/2 is thus related to the densitybykF = (4ns/gsgv)

1/2. The second subband starts tobe populated when EF exceeds the energy of the secondband bottom. The stepwise increasing density of statesshown in Fig. 6a is referred to as quasi-two-dimensional.As the number of occupied subbands increases, the den-sity of states eventually approaches the

Edependence

characteristic for a three-dimensional system. Note, how-ever, that usually only a single subband is occupied.

If the 2DEG is confined laterally to a narrow channel,then Eq. (1.2) only represents the kinetic energy from thefree motion (with momentum hk)parallelto the channelaxis. Because of the lateral confinement, a single two-dimensional (2D) subband is split itself into a series ofone-dimensional (1D) subbands, with band bottoms atEn, n = 1, 2, . . . The total energy En(k) of an electronin thenth 1D subband (relative to the bottom of the 2Dsubband) is given by

En(k) =En+ h2k2/2m. (1.4)

Two frequently used potentials to model analytically

the lateral confinement are the square-well potential (ofwidthW, illustrated in Fig.6b) and the parabolic poten-tial well (described by V(x) = 12m

20x

2). The confine-ment levels are then given either byEn = (nh)

2/2mW2

for the square well or by En = (n 12)h0 for theparabolic well. When one considers electron transportthrough a narrow channel, it is useful to distinguish be-tween states with positive and negative k, since thesestates move in opposite directions along the channel. Wedenote by+n (E) the density of states withk >0 per unitchannel length in the nth 1D subband. This quantity isgiven by

+n (E) = gsgv

2dE

n(k)

dk1

= gsgvm

2h2

h2

2m(E En)1/2

. (1.5)

The density of states n with k < 0 is identical to +n .

(This identity holds because of time-reversal symmetry;In a magnetic field, +n = n , in general.) The totaldensity of states (E), drawn in Fig. 6b, is twice theresult (1.5) summed over all n for which En < E. The


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FIG. 6 Density of states (E) as a function of energy. (a)Quasi-2D density of states, with only the lowest subband oc-cupied (hatched). Inset: Confinement p otential perpendicu-lar to the plane of the 2DEG. The discrete energy levels corre-spond to the bottoms of the first and second 2D subbands. (b)Quasi-1D density of states, with four 1D subbands occupied.Inset: Square-well lateral confinement potential with discreteenergy levels indicating the 1D subband bottoms. (c) Densityof states for a 2DEG in a perpendicular magnetic field. Theoccupied 0D subbands or Landau levels are shown in black.Impurity scattering may broaden the Landau levels, leadingto a nonzero density of states between the peaks.

density of states of a quasi-one-dimensional electron gaswith many occupied 1D subbands may be approximatedby the 2D result (1.3).

If a magnetic field B is applied perpendicular to anunbounded 2DEG, the energy spectrum of the electronsbecomes fully discrete, since free translational motion inthe plane of the 2DEG is impeded by the Lorentz force.Quantization of the circular cyclotron motion leads to

energy levels at93

En= (n 12)hc, (1.6)withc = eB/m the cyclotron frequency. The quantumnumber n = 1, 2, . . .labels the Landau levels. The num-ber of states is the same in each Landau level and equalto one state (for each spin and valley) per flux quantumh/e through the sample. To the extent that broadeningof the Landau levels by disorder can be neglected, thedensity of states (per unit area) can be approximated by

(E) = gs

gv

eB

h

n=1

(E

En

), (1.7)

as illustrated in Fig.6c. The spin degeneracy containedin Eq. (1.7) is resolved in strong magnetic fields as a re-sult of the Zeeman splitting gBB of the Landau levels(B eh/2me denotes the Bohr magneton; the Landeg-factor is a complicated function of the magnetic fieldin these systems).20 Again, if a large number of Landaulevels is occupied (i.e., at weak magnetic fields), one re-covers approximately the 2D result (1.3). The foregoingconsiderations are for an unbounded 2DEG. A magneticfield perpendicular to a narrow 2DEG channel causes thedensity of states to evolve gradually from the 1D form of

Fig.6b to the effectively 0D form of Fig. 6c. This tran-sition is discussed in SectionII.F.

2. Drude conductivity, Einstein relation, and Landauer formula

In the presence of an electric field E in the plane ofthe 2DEG, an electron acquires a drift velocity v =eEt/m in the time t since the last impurity col-lision. The average of tis the scattering time , so theaverage drift velocityvdrift is given by

vdrift = eE, e = e/m. (1.8)

The electron mobility e together with the sheet den-sity ns determine the conductivity in the relationensvdrift = E. The result is the familiar Drudeconductivity,94 which can be written in several equiva-lent forms:

= ense =e2ns

m =gsgv

e2

h

kFl

2 . (1.9)

In the last equality we have used the identity ns =gsgvk

2F/4(see SectionI.D.1) and have defined the mean


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free path l = vF. The dimensionless quantity kFl ismuch greater than unity in metallic systems (see Table Ifor typical values in a 2DEG), so the conductivity is largecompared with the quantum unite2/h (26k)1.

From the preceding discussion it is obvious that thecurrent induced by the applied electric field is carried byallconduction electrons, since each electron acquires thesame average drift velocity. Nonetheless, to determine

the conductivity it is sufficient to consider the responseof electronsnear the Fermi levelto the electric field. Thereason is that the states that are more than a few timesthe thermal energykBTbelowEF are all filled so that inresponse to a weak electric field only the distribution ofelectrons among states at energies close toEF is changedfrom the equilibrium Fermi-Dirac distribution

f(E EF) =

1 + expE EF

kBT

1. (1.10)

The Einstein relation94

= e2(EF)D (1.11)

is one relation between the conductivity and Fermi levelproperties (in this case the density of states (E) andthe diffusion constant D, both evaluated at EF). TheLandauer formula4 [Eq. (1.22)] is another such relation(in terms of the transmission probability at the Fermilevel rather than in terms of the diffusion constant).

The Einstein relation (1.11) for an electron gas at zerotemperature follows on requiring that the sum of the driftcurrent densityE/e and the diffusion current densityDns vanishes in thermodynamic equilibrium, charac-terized by a spatially constant electrochemical potential:

E/e Dns = 0, when = 0. (1.12)The electrochemical potential is the sum of the electro-static potential energyeV (which determines the en-ergy of the bottom of the conduction band) and the chem-ical potential EF (being the Fermi energy relative to theconduction band bottom). Since (at zero temperature)dEF/dns = 1/(EF), one has

= eE + (EF)1ns. (1.13)The combination of Eqs. (1.12) and (1.13) yields the Ein-stein relation (1.11) between and D. To verify that Eq.

(1.11) is consistent with the earlier expression (1.9) forthe Drude conductivity, one can use the result (see be-low) for the 2D diffusion constant:

D= 12v2F=

12

vFl, (1.14)

in combination with Eq. (1.3) for the 2D density of states.At a finite temperature T, a chemical potential (or

Fermi energy) gradientEF induces a diffusion currentthat is smeared out over an energy range of order kBTaround EF. The energy interval between Eand E+ dE

contributes to the diffusion current density j an amountdj given by

djdiff = D{(E)f(E EF)dE}= dED(E) df

dEFEF, (1.15)

where the diffusion constantD is to be evaluated at en-ergy E. The total diffusion current density follows onintegration overE:

j= EFe2 0

dE (E, 0) df

dEF, (1.16)

with(E, 0) the conductivity (1.11) at temperature zerofor a Fermi energy equal to E. The requirement of van-ishing current for a spatially constant electrochemical po-tential implies that the conductivity(EF, T) at temper-atureTand Fermi energy EF satisfies

(EF, T)e2EF+j= 0.

Therefore, the finite-temperature conductivity is given

simply by the energy average of the zero-temperatureresult

(EF, T) =

0

dE (E, 0) df

dEF. (1.17)

AsT 0, df/dEF (E EF), so indeed onlyE= EFcontributes to the energy average. Result(1.17) containsexclusively the effects of a finite temperature that are dueto the thermal smearing of the Fermi-Dirac distribution.A possible temperature dependence of the scattering pro-cesses is not taken into account.

We now want to discuss one convenient way to calcu-late the diffusion constant (and hence obtain the conduc-

tivity). Consider the diffusion current densityjx due toa small constant density gradient, n(x) = n0+ cx. Wewrite

jx = limt

vx(t= 0)n(x(t= t))= lim

tcvx(0)x(t)

= limt

c t0

dtvx(0)vx(t), (1.18)

wheretis time and the brackets denote an isotropicangular average over the Fermi surface. The time intervalt , so the velocity of the electron at time 0 isuncorrelated with its velocity at the earlier timet.This allows us to neglect at x(t) the small deviationsfrom an isotropic velocity distribution induced by thedensity gradient [which could not have been neglectedat x(0)]. Since only the time difference matters in thevelocity correlation function, one hasvx(0)vx(t) =vx(t)vx(0). We thus obtain for the diffusion constantD= jx/cthe familiar linear response formula95

D=

0

dtvx(t)vx(0). (1.19)


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10

Since, in the semiclassical relaxation time approximation,each scattering event is assumed to destroy all correla-tions in the velocity, and since a fraction exp(t/) ofthe electrons has not been scattered in a time t, one has(in 2D)

vx(t)vx(0) = vx(0)2et/ = 12v2Fet/. (1.20)Substituting this correlation function for the integrandin Eq. (1.19), one recovers on integration the diffusionconstant (1.14).

The Drude conductivity (4.8) is a semiclassical result,in the sense that while the quantum mechanical Fermi-Dirac statistic is taken into account, the dynamics of theelectrons at the Fermi level is assumed to be classical.In Section II we will discuss corrections to this resultthat follow from correlations in the diffusion process dueto quantum interference. Whereas for classical diffusioncorrelations disappear on the time scale of the scatteringtime [as expressed by the correlation function (1.20)],in quantum diffusion correlations persist up to times ofthe order of the phase coherence time. The latter time

is associated with inelastic scattering and at low tem-peratures can become much greater than the time as-sociated with elastic scattering.

In an experiment one measures a conductance ratherthan a conductivity. The conductivity relates the localcurrent density to the electric field, j = E, while theconductance G relates the total current to the voltagedrop, I = GV. For a large homogeneous conductor thedifference between the two is not essential, since Ohmslaw tells us that

G= (W/L) (1.21)

for a 2DEG of width W and length L in the current

direction. (Note that G and have the same units intwo dimensions.) If for the moment we disregard the ef-fects of phase coherence, then the simple scaling (1.21)holds provided bothW and L are much larger than themean free path l . This is the diffusive transport regime,illustrated in Fig.7a. When the dimensions of the sam-ple are reduced below the mean free path, one enterstheballistictransport regime, shown in Fig.7c. One canfurther distinguish an intermediatequasi-ballisticregime,characterized by W < l < L(see Fig. 7b). In ballistictransport only the conductance plays a role, not the con-ductivity. The Landauer formula

G= (e

2

/h)T (1.22)plays a central role in the study of ballistic transport be-cause it expresses the conductance in terms of a Fermilevel property of the sample (the transmission probabil-ityT, see SectionIII.A). Equation (1.22) can thereforebe applied to situations where the conductivity does notexist as a local quantity, as we will discuss in Sections IIIandIV.

If phase coherence is taken into account, then the mini-mal length scale required to characterize the conductivity

FIG. 7 Electron trajectories characteristic for the diffu-

sive (l < W , L), quasi-ballistic (W < l < L), and ballistic(W,L < l) transport regimes, for the case of specular bound-ary scattering. Boundary scattering and internal impurityscattering (asterisks) are of equal importance in the quasi-ballistic regime. A nonzero resistance in the ballistic regimeresults from back scattering at the connection between thenarrow channel and the wide 2DEG regions. Taken fromH. van Houten et al., in Physics and Technology of Submi-cron Structures(H. Heinrich, G. Bauer, and F. Kuchar, eds.).Springer, Berlin, 1988.

becomes larger. Instead of the (elastic) mean free pathl vF, the phase coherence length l (D)1/2 be-comes this characteristic length scale (up to a numericalcoefficientlequals the average distance that an electrondiffuses in the time ). Ohms law can now only be ap-plied to add the conductances of parts of the sample withdimensions greater thanl. Since at low temperatureslcan become quite large (cf. TableI), it becomes possiblethat (for a small conductor) phase coherence extends overa large part of the sample. Then only the conductance(not the conductivity) plays a role, even if the transportis fully in the diffusive regime. We will encounter suchsituations repeatedly in SectionII.

3. Magnetotransport

In a magnetic field B perpendicular to the 2DEG, thecurrent is no longer in the direction of the electric fielddue to the Lorentz force. Consequently, the conductiv-ity is no longer a scalar but a tensor , related via theEinstein relation = e2(EF)Dto the diffusion tensor

D=

0

dtv(t)v(0). (1.23)


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11

Equation (1.23) follows from a straightforward general-ization of the argument leading to the scalar relation(1.19) [but now the ordering ofv(t) and v(0) matters].Between scattering events the electrons at the Fermilevel execute circular orbits, with cyclotron frequencyc = eB/m and cyclotron radius lcycl =mvF/eB. Tak-ing the 2DEG in thexyplane, and the magnetic field inthe positivez-direction, one can write in complex number

notation

v(t) vx(t) + ivy(t) = vFexp(i + ict). (1.24)The diffusion tensor is obtained from

Dxx+ iDyx =

20

d

2

0

dt v(t)vFcos et/

= D

1 + (c)2(1 + ic), (1.25)

where D is the zero-field diffusion constant (1.14). Oneeasily verifies that Dyy = Dxx and Dxy =Dyx. Fromthe Einstein relation one then obtains the conductivity

tensor

=

1 + (c)2

1 cc 1

, (1.26)

with the zero-field conductivity (1.9). The resistivitytensor 1 has the form

=

1 c

c 1

, (1.27)

with = 1 =m/nse2the zero-field resistivity.The off-diagonal elementxy RHis the classicalHall

resistance of a 2DEG:

RH = B

nse=

1

gsgv

h

e2hcEF

. (1.28)

Note that in a 2D channel geometry there is no distinc-tion between the Hall resistivityand the Hall resistance,since the ratio of the Hall voltage VH = W Ex across thechannel to the current I= W jy along the channel doesnot depend on its length and width (provided transportremains in the diffusive regime). The diagonal elementxx is referred to as the longitudinal resistivity. Equa-tion (1.27) tells us that classically the magnetoresistivityis zero (i.e., xx(B) xx(0) = 0). This counterintuitiveresult can be understood by considering that the forcefrom the Hall voltage cancels the average Lorentz forceon the electrons. A general conclusion that one can drawfrom Eqs. (1.26) and (1.27) is that the classical effects ofa magnetic field are important only ifc > 1. In suchfields an electron can complete several cyclotron orbitsbefore being scattered out of orbit. In a high-mobility2DEG this criterion is met at rather weak magnetic fields(note thatc=eB, and see TableI).

In the foregoing application of the Einstein relation wehave used the zero-field density of states. Moreover, we

FIG. 8 Schematic dependence on the reciprocal filling factor1 2eB/hns of the longitudinal resistivity xx (normal-ized to the zero-field resistivity ) and of the Hall resistance

RH xy (normalized to h/2e2). The plot is for the caseof a single valley with twofold spin degeneracy. Deviationsfrom the semiclassical result (1.27) occur in strong magneticfields, in the form of Shubnikov-De Haas oscillations in xxand quantized plateaus [Eq. (1.31)] in xy.

have assumed that the scattering time isB-independent.Both assumptions are justified in weak magnetic fields,for which EF/hc 1, but not in stronger fields (cf.Table I). As illustrated in Fig. 8, deviations from thesemiclassical result (1.27) appear as the magnetic field isincreased. These deviations take the form of an oscilla-

tory magnetoresistivity (the Shubnikov-De Haas effect)and plateaux in the Hall resistance (the quantum Halleffect). The origin of these two phenomena is the forma-tion of Landau levels by a magnetic field, discussed inSectionI.D.1, that leads to the B-dependent density ofstates (1.7). The main effect is on the scattering rate 1,which in a simple (Born) approximation96 is proportionalto(EF):

1 = (/h)(EF)ciu2. (1.29)

Here ci is the areal density of impurities, and the im-purity potential is modeled by a 2D delta function ofstrengthu. The diagonal element of the resistivity ten-sor (1.27) is xx= (m/e2ns)1 (EF). Oscillations inthe density of states at the Fermi level due to the Landaulevel quantization are therefore observable as an oscilla-tory magnetoresistivity. One expects the resistivity to beminimal when the Fermi level lies between two Landaulevels, where the density of states is smallest. In view ofEq. (1.7), this occurs when the Landau level filling fac-tor (ns/gsgv)(h/eB) equals an integer N = 1, 2, . . .(assuming spin-degenerate Landau levels). The resultingShubnikov-De Haas oscillations are periodic in 1/B, with


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12

spacing (1/B) given by

(1

B) =

e

h

gsgvns

, (1.30)

providing a means to determine the electron density froma magnetoresistance measurement. This brief explana-tion of the Shubnikov-De Haas effect needs refinement,20

but is basically correct. The quantum Hall effect,8 being

the occurrence of plateaux in RH versusB at precisely

RH = 1

gsgv

h

e21

N, N= 1, 2, . . . , (1.31)

is a more subtle effect97 to which we cannot do justicein a few lines (see Section IV.A). The quantization ofthe Hall resistance is related on a fundamental level tothe quantization in zero magnetic field of the resistanceof a ballistic point contact.6,7 We will present a unifieddescription of both these effects in Sections III.A andIII.B.

II. DIFFUSIVE AND QUASI-BALLISTIC TRANSPORT

A. Classical size effects

In metals, the dependence of the resistivity on thesize of the sample has been the subject of study foralmost a century.98 Because of the small Fermi wavelength in a metal, these are classical size effects. Com-prehensive reviews of this field have been given byChambers,99 Brandli and Olsen,100 Sondheimer,101 and,recently, Pippard.102 In semiconductor nanostructuresboth classical and quantum size effects appear, and anunderstanding of the former is necessary to distinguishthem from the latter. Classical size effects in a 2DEG are

of intrinsic interest as well. First of all, a 2DEG is an idealmodel system to study known size effects without thecomplications of nonspherical Fermi surfaces and poly-crystallinity, characteristic for metals. Furthermore, it ispossible in a 2DEG to study the case of nearly completespecular boundary scattering, whereas in a metal diffusescattering dominates. The much smaller cyclotron radiusin a 2DEG, compared with a metal at the same magneticfield value, allows one to enter the regime where the cy-clotron radius is comparable to the range of the scatteringpotential. The resulting modifications of known effects inthe quasi-ballistic transport regime are the subject of thissection. A variety of new classical size effects, not knownfrom metals, appear in the ballistic regime, when the re-sistance is measured on a length scale below the meanfree path. These are discussed in SectionIII.E, and re-quire a reconsideration of what is meant by a resistanceon such a short length scale.

In the present section we assume that the channellengthL (or, more generally, the separation between thevoltage probes) is much larger than the mean free path lfor impurity scattering so that the motion remains diffu-sive along the channel. Size effects in the resistivity oc-cur when the motion across the channel becomes ballistic

(i.e., when the channel width W < l). Diffuse bound-ary scattering leads to an increase in the resistivity in azero magnetic field and to a nonmonotonic magnetore-sistivity in a perpendicular magnetic field, as discussedin the following two subsections. The 2D channel ge-ometry is essentially equivalent to the 3D geometry ofa thin metal plate in a parallel magnetic field, with thecurrent flowing perpendicular to the field. Size effects in

this geometry were originally studied by Fuchs103

in azero magnetic field and by MacDonald104 for a nonzerofield. The alternative configuration in which the mag-netic field is perpendicular to the thin plate, studied bySondheimer105 does not have a 2D analog. We discussin this section only the classical size effects, and thus thediscreteness of the 1D subbands and of the Landau lev-els is ignored. Quantum size effects in the quasi-ballistictransport regime are treated in SectionII.F.

1. Boundary scattering

In a zero magnetic field, scattering at the channelboundaries increases the resistivity, unless the scatteringis specular. Specular scattering occurs if the confiningpotential V(x, y) does not depend on the coordinate yalong the channel axis. In that case the electron motionalong the channel is not influenced at all by the lateralconfinement, so the resistivity retains its 2D bulk value0 = m/e2ns. More generally, specular scattering re-quires any roughness of the boundaries to be on a lengthscale smaller than the Fermi wavelength F. The con-fining potential created electrostatically by means of agate electrode is known to cause predominantly specu-lar scattering (as has been demonstrated by the electron

focusing experiments

59

discussed in SectionIII.C). Thisis a unique situation, not previously encountered in met-als, where as a result of the small F (on the order ofthe interatomic separation) diffuse boundary scatteringdominates.102

Diffusescattering means that the velocity distributionat the boundary is isotropic for velocity directions thatpoint away from the boundary. Note that this impliesthat an incident electron is reflected with a (normal-ized) angular distribution P() = 12cos , since the re-flection probability is proportional to the flux normal tothe boundary. Diffuse scattering increases the resistivityabove0 by providing an upper bound Wto the effectivemean free path. In order of magnitude,

(l/W)0 if

l > W (a more precise expression is derived later). Ingeneral, boundary scattering is neither fully specular norfully diffuse and, moreover, depends on the angle of inci-dence (grazing incidence favors specular scattering sincethe momentum along the channel is large and not eas-ily reversed). The angular dependence is often ignoredfor simplicity, and the boundary scattering is described,following Fuchs,103 by a single parameter p, such thatan electron colliding with the boundary is reflected spec-ularly with probability p and diffusely with probability


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13

1 p. This specularity parameter is then used as a fitparameter in comparison with experiments. Soffer106 hasdeveloped a more accurate, and more complicated, mod-eling in terms of an angle of incidence dependent specu-larity parameter.

In the extreme case of fully diffuse boundary scatter-ing (p= 0), one is justified in neglecting the dependenceof the scattering probability on the angle of incidence.

We treat this case here in some detail to contrast it withfully specular scattering, and because diffuse scatteringcan be of importance in 2DEG channels defined by ionbeam exposure rather than by gates.107,108 We calculatethe resistivity from the diffusion constant by means ofthe Einstein relation. Fuchs takes the alternative (butequivalent) approach of calculating the resistivity fromthe linear response to an applied electric field.103 Im-purity scattering is taken as isotropic and elastic and isdescribed by a scattering time such that an electronis scattered in a time interval dt with probability dt/,regardless of its position and velocity, This is the com-monly employed scattering time (or relaxation time)

approximation.The channel geometry is defined by hard walls at

x = W/2 at which the electrons are scattered dif-fusely. The stationary electron distribution function atthe Fermi energy F(r, ) satisfies the Boltzmann equa-tion

v r

F = 1

F+1

20

d

2F, (2.1)

where r (x, y) is the position and is the angle thatthe velocity v vF(cos , sin ) makes with the x-axis.The boundary condition corresponding to diffuse scat-

tering is that Fis independent of the velocity directionfor velocities pointing away from the boundary. In viewof current conservation this boundary condition can bewritten as

F(r, ) = 1

2

/2/2

d F(r, )cos ,

for x=W

2 ,

2 < 0a

lT, l L l L, lT lT l L

G 2gsgv

1/2 Ce2

h C

e2

h

lL

3/2C

e2

h

lTl1/2

L3/2

C 0.73

12

8

3

1/2aThe results assume a narrow channel (W L), with a 2D den-

sity of states (W F), which is in the 1D limit for the con-ductance fluctuations (W l). The expressions for G are fromRefs.140,141,145, and146. The numerical prefactor C for T = 0 isfrom Ref.141, for T > 0 from Ref.147. If time-reversal symmetryapplies, then = 1, but in the presence of a magnetic field strongenough to suppress the cooperon contributions then = 2. If the

spin degeneracy is lifted, gs is to be replaced by g1/2s .

F(0)1/2. The correlation field Bc is defined as the half-

width at half-height F(Bc) F(0)/2. The correlationfunctionF(B) is determined theoretically141,145,146 bytemporal and spatial integrals of two propagators: thediffuson Pd(r, r, t) and the cooperon Pc(r, r, t). As dis-cussed by Chakravarty and Schmid,126 these propagatorsconsist of the product of three terms: (1) the classicalprobability to diffuse from r to r in a time t (indepen-dent ofB in the field rangec 1 of interest here); (2)the relaxation factor exp(t/), which describes the lossof phase coherence due to inelastic scattering events; (3)the average phase factor exp(i), which describes theloss of phase coherence due to the magnetic field. Theaverage is taken over all classical trajectories thatdiffuse fromr to r in a time t. The phase difference is different for a diffuson or cooperon:

(diffuson) = e

h

rr

A dl, (2.40a)

(cooperon) = e

h

r

r

(2A + A) dl, (2.40b)

where the line integral is along a classical trajectory.The vector potentialA corresponds to the magnetic fieldB = A, and the vector potential increment Acorresponds to the field increment B in the correlationfunctionF(B) (according to B= A). An ex-planation of the different magnetic field dependencies ofthe diffuson and cooperon in terms of Feynman paths isgiven shortly.

In Ref.109 we have proven that in a narrow channel(W l) the average phase factorexp(i) does notdepend on initial and final coordinates r and r, pro-vided that one works in the Landau gauge and thatt . This is a very useful property, since it allowsone to transpose the results forexp(i) obtained forr= r in the context of weak localization to the presentproblem of the conductance fluctuations, where r can be

different from r. We recall that for weak localization

the phase difference is that of the cooperon, withthe vector potential increment A = 0 [cf. Eq. (2.15)].The average phase factor then decays exponentially asexp(i) = exp(t/B) [cf. Eq. (2.21)], with the re-laxation time B given as a function of magnetic field Bin TableII. We conclude that the same exponential decayholds for the average cooperon and diffuson phase factorsafter substitution ofB B+ B/2 and B B/2,respectively, in the expressions for B :

ei(diffuson) = exp(t/B/2), (2.41a)ei(cooperon) = exp(t/B+B/2). (2.41b)

The cooperon is suppressed when B+B/2 < ,which occurs on the same field scale as the suppression

of weak localization (determined by B Bc (i.e., for fields beyond the weaklocalization peak). Then only the diffuson contributes tothe conductance fluctuations, since the relaxation time ofthe diffuson is determined by the field increment B inthe correlation function F(B), not by the magnetic fielditself. This is the critical difference with weak localiza-tion: The conductance fluctuations arenot suppressedbya weak magnetic field. The different behavior of cooper-ons and diffusons can be understood in terms of Feyn-man paths. The correlation function F(B) containsthe product of four Feynman path amplitudes A(i, B),A(j, B), A(k, B+ B), and A(l, B+ B) along var-


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27

FIG. 21 Illustration of the different flux sensitivity of theinterference terms of diffuson type (a) and of cooperon type(b). Both contribute to the conductance fluctuations in a zeromagnetic field, but the cooperons are suppressed by a weakmagnetic field, as discussed in the text.

ious paths i ,j,k,l from r to r. Consider the diffusonterm for which i= l and j = k . The phase of this termA(i, B)A(j, B)A(j, B+ B)A(i, B+ B) is

e

h A dl +

e

h (A + A) dl=

e

h. (2.42)

where the line integral is taken along the closed loopformed by the two paths i and j (cf. Fig. 21a). Thephase is thus given by the flux increment SBthrough this loop and does not contain the flux SBitself. The fact that the magnetic relaxation time of thediffuson depends only on B and not on B is a conse-quence of the cancellation contained in Eq. (2.42). Forthe cooperon, the relevant phase is that of the productof Feynman path amplitudesA(i, B)A(j, B)A+(j, B +B)A+(i, B+ B), where thesign refers to a trajec-tory from r to r and the + sign to a trajectory from rtor (see Fig.21b). This phase is given by

e

h

A dl + e

h

(A + A) dl= e

h(2 + ). (2.43)

In contrast to the diffuson, the cooperon is sensitive tothe flux through the loop and can therefore be sup-pressed by a weak magnetic field.

In the following, we assume that B > Bc so that onlythe diffuson contributes to the magnetoconductance fluc-tuations. The combined effects of magnetic field and in-elastic scattering lead to a relaxation rate

1eff =1 +

1B/2

, (2.44)

which describes the exponential decay of the averagephase factor ei = exp(t/eff). Equation (2.44) con-tains the whole effect of the magnetic field on the dif-fuson. Without having to do any diagrammatic analy-sis, we therefore conclude147 that the correlation func-tion F(B) can be obtained from the variance F(0)Var G= (G)2 (given in TableIII) by simply replacingby the effective relaxation time effdefined in Eq. (2.44).The quantity B/2 corresponds to the magnetic relax-ation time B obtained for weak localization (see Table

II) after substitution ofBB/2. For easy reference,we give the results for the dirty and clean metal regimesexplicitly:109,147

B/2 = 12

h

eB

21

DW2, if l W, (2.45)

B/2 = 4C1 h

eB

21

vFW3

+ 2C2 h

eB

l

vFW2

,

if l W, (2.46)where C1 = 9.5 and C2 = 24/5 for a channel with spec-ular boundary scattering (C1 = 4 and C2 = 3 for achannel with diffuse boundary scattering). These resultsare valid under the condition W2B h/e, which fol-lows from the requirement eff that the electronicmotion on the effective phase coherence time scale effbe diffusive rather than ballistic, as well as from the re-quirement (Deff)1/2 Wfor one-dimensionality.

With results (2.44)(2.46), the equation F(Bc) =F(0)/2, which defines the correlation field Bc, reducesto an algebraic equation that can be solved straightfor-

wardly. In the dirty metal regime one finds145

Bc = 2Ch

e

1

W l, (2.47)

where the prefactorCdecreases from147 0.95 forl lTto 0.42 for l lT. Note the similarity with the result(2.19) for weak localization. Just as in weak localiza-tion, one finds that the correlation field in the pure metalregime is significantly enhanced above Eq. (2.47) due tothe flux cancellation effect discussed in Section II.B.3.The enhancement factor increases from (l/W)1/2 tol/Was ldecreases from above to below the length l

3/2W1/2.

The relevant expression is given in Ref.147

. As an illus-tration, the dimensionless correlation flux BcW le/hinthe pure and dirty metal regimes is plotted as a functionofl/l in Fig.22 for lT l.

In the following discussion of the experimental situ-ation in semiconductor nanostructures, it is importantto keep in mind that the Altshuler-Lee-Stone theory ofconductance fluctuations was formulated for an appli-cation to metals. This has justified the neglect of sev-eral possible complications, which may be important ina 2DEG. One of these is the classical curvature of theelectron trajectories, which affects the conductance whenlcycl


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28

FIG. 22 Plot of the dimensionless correlation flux c BclWe/h for the magnetoconductance fluctuations as afunction ofl/l in the regime lT l. The solid curve is forthe case l = 5 W; the dashed line is for l W. Taken fromC. W. J. Beenakker and H. van Houten, Phys. Rev. B 37,

6544 (1988).

In the following discussion of experimental studies ofconductance fluctuations, we will have occasion to discussbriefly one further development. This is the modificationof the theory149,150,151,152,153,154 to account for the dif-ferences between two- and four-terminal measurementsof the conductance fluctuations, which becomes impor-tant when the voltage probes are separated by less thanthe phase coherence length.155,156

4. Experiments

The experimental observation of conductance fluctua-tions in semiconductors has preceded the theoretical un-derstanding of this phenomenon. Weak irregular con-ductance fluctuations in wide Si inversion layers were re-ported in 1965 by Howard and Fang.157 More pronouncedfluctuations were found by Fowler et al. in narrow Si accu-mulation layers in the strongly localized regime.32 Kwas-nick et al. made similar observations in narrow Si inver-sion layers in the metallic conduction regime.39 Thesefluctuations in the conductance as a function of gatevoltage or magnetic field have been tentatively explainedby various mechanisms.158 One of the explanations sug-gested is based on resonant tunneling,159 another on vari-able range hopping. At the 1984 conference on Elec-tronic Properties of Two-Dimensional Systems Wheeleret al.161 and Skocpol et al.162 reported pronounced struc-ture as a function of gate voltage in the low-temperatureconductance of narrow Si inversion layers, observed inthe course of their search for a quantum size effect.

After the publication in 1985 of the Altshuler-Lee-Stone theory140,141,163 of universal conductance fluctua-tions, a consensus has rapidly developed that this theory

FIG. 23 Negative magnetoresistance and aperiodic magne-toresistance fluctuations in a narrow Si inversion layer channelfor several values of the gate voltage VG. Note that the verti-cal offset and scale is different for each VG. Taken from J. C.Licini et al., Phys. Rev. Lett. 55, 2987 (1985).

properly accounts for the conductance fluctuations in themetallic regime, up to factor of two uncertainties in thequantitative description.46,144,164 Following this theoreti-cal work, Licini et al.40 attributed the magnetoresistanceoscillations that they observed in narrow Si inversion lay-ers to quantum interference in a disordered conductor.Their low-temperature measurements, which we repro-duce in Fig.23, show a large negative magnetoresistancepeak due to weak localization at low magnetic fields, inaddition to aperiodic fluctuations that persist to highfields. Such a clear weak localization peak is not found in

shorter samples, where the conductance fluctuations arelarger. The reason is that the magnitude of the conduc-tance fluctuations G is proportional to (l/L)

3/2 [forl lT, cf. Eq. (2.34)], while the weak localization con-ductance correction scales with l/L [as discussed belowEq. (2.14)]. Weak localization thus predominates in longchannels (L l) where the fluctuations are relativelyunimportant.

The most extensive quantitative study of the univer-sality of the conductance fluctuations in narrow Si inver-sion layers (over a wide range of channel widths, lengths,gate voltages, and temperatures) was made by Skocpol etal.45,46,156 In the following, we review some of these ex-perimental results. We will not discuss the similarly ex-tensive investigations by Webb et al.155,164,165 on smallmetallic samples, which have played an equally impor-tant role in the development of this subject. To analyzetheir experiments, Skocpol et al. estimated l from weaklocalization experiments (with an estimated uncertaintyof about a factor of 2). They then plotted the root-mean-square variation G of the conductance as a function ofL/l, with L the separation of the voltage probes in thechannel. Their results are shown in Fig.24. The pointsforL > l convincingy exhibit for a large variety of data


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sets the (L/l)3/2 scaling law predicted by the theory

described in SectionII.C.3(for l< lT, which is usuallythe case in Si inversion layers).

For L < l the experimental data of Fig. 24show acrossover to a (L/l)

2 scaling law (dashed line), ac-companied by an increase of the magnitude of the con-ductance fluctuations beyond the value G e2/h pre-dicted by the Altshuler-Lee-Stone theory for a conduc-

tor of length L < l. A similar observation was madeby Benoit et al.155 on metallic samples. The disagree-ment is explained155,156 by considering that the experi-mental geometry differs from that assumed in the the-ory discussed in Section II.C.3. Use is made of a longchannel with voltage probes at different spacings. Theexperimental L is the spacing of two voltage probes,and not the length of a channel connecting two phase-randomizing reservoirs, as envisaged theoretically. Thedifference is irrelevant if L > l. If the probe separa-tion L is less than the phase coherence length l, how-ever, the measurement still probes a channel segmentof length l rather than L. In this sense the measure-

ment is nonlocal.155,156

The key to the L2

dependenceof G found experimentally is that the voltages on theprobes fluctuate independently, implying that the resis-tancefluctuationsR are independent ofLin this regimeso that G R2R L2. This explanation is con-sistent with the anomalously small correlation field Bcfound for L < l.46,156 One might have expected thatthe result Bc h/eWl for L > l should be replacedby the larger value Bc h/eWL ifL is reduced belowl. The smaller value found experimentally is due tothe fact that the flux through parts of the channel adja-cent to the segment between the voltage probes, as wellas the probes themselves, has to be taken into account.These qualitative arguments155,156 are supported by de-tailed theoretical investigations.149,150,151,152,153,154 Theimportant message of these theories and experiments isthat the transport in a small conductor is phase coherentover large length scales and that phase randomization(due to inelastic collisions) occurs mainly as a result ofthe voltage probes. The Landauer-Buttiker formalism4,5

(which we will discuss in SectionIII.A) is naturally suitedto study such problems theoretically. In that formal-ism, current and voltage contacts are modeled by phase-randomizing reservoirs attached to the conductor. We re-fer to a paper by Buttiker149 for an instructive discussionof conductance fluctuations in a multiprobe conductor interms of interfering Feynman paths.

Conductance fluctuations have also been observedin narrow-channel GaAs-AlGaAs heterostructures.166,167

These systems are well in the pure metal regime (W < l),but unfortunately they are only marginally in the regimeof coherent diffusion (characterized by ). Thishampers a quantitative comparison with the theoreticalresults147 for the pure metal regime discussed in SectionII.C.3. (A phenomenological treatment of conductancefluctuations in the case that is given in Refs.168and169.) The data of Ref.167 are consistent with an en-

FIG. 24 Root-mean-square amplitudeg of the conductancefluctuations (in units ofe2/h) as a function of the ratio of the

distance between the voltage probes L to the estimated phasecoherence length l for a set of Si inversion layer channelsunder widely varying experimental conditions. The solid anddashed lines demonstrate the (L/l)

3/2 and (L/l)2 scaling

ofg in the regimes L > l and L < l, respectively. Takenfrom W. J. Skocpol, Physica Scripta T19, 95 (1987).

hancement of the correlation field due to the flux cancel-lation effect, but are not conclusive.147 We note that theflux cancellation effect can also explain the correlationfield enhancement noticed in a computer simulation byStone.163

In the analysis of the aforementioned experiments onmagnetoconductance fluctuations, a twofold spin degen-eracy has been assumed. The variance (G)2 is reducedby a factor of 2 if the spin degeneracy is lifted by astrong magnetic field B > Bc2. The Zeeman energygBB should be sufficiently large than the spin-up andspin-down electrons give statistically independent contri-butions to the conductance. The degeneracy factor g2s in(G)2 (introduced in Section II.C.1) should then be re-placed by a factor gs, since the variances of statisticallyindependent quantities add. Sincegs = 2, one obtains afactor-of-2 reduction in (G)2. Note that this reductioncomes on top of the factor-of-2 reduction in (G)2 due tothe breaking of time-reversal symmetry, which occurs at

weak magnetic fieldsBc. Stone has calculated170 that thefield Bc2 in a narrow channel (l W) is given by thecriterion of unit phase changeg BB/h in a coherencetime, resulting in the estimateBc2 h/gB. Surpris-ingy, the thermal energy kBTis irrelevant for Bc2 in the1D casel W(but not in higher dimensions170).

For the narrow-channel experiment of Ref.167 just dis-cussed, one finds (using the estimates 7ps andg 0.4) a crossover field Bc2 of about 2 T, well abovethe field range used for the data analysis.147 Most im-


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portantly, no magnetoconductance fluctuations are ob-served if the magnetic field is applied parallel to the2DEG (see Section II.E), demonstrating that the Zee-man splitting has no effect on the conductance in thisfield regime. More recently, Debray et al.171 performedan experimental study of the reduction by a perpendic-ular magnetic field of the conductance fluctuations as afunction of Fermi energy (varied by means of a gate).

The estimated value of is larger than that of Ref.167

by more than an order of magnitude. Consequently, avery small Bc2 0.07 T is estimated in this experiment.The channel is relatively wide (2 m lithographic width),so the field Bc for time-reversal symmetry breaking iseven smaller (Bc 7 104 T). A total factor-of-4 re-duction in (G)2 was found, as expected. The valuesof the observed crossover fields Bc and Bc2 also agreereasonably well with the theoretical prediction. Unfor-tunately, the magnetoconductance in a parallel magneticfield was not investigated by these authors, which wouldhave provided a definitive test for the effect of Zeemansplitting on the conductance aboveBc2. We note that re-lated experimental172,173 and theoretical174,175 work hasbeen done on the reduction oftemporalconductance fluc-tuations by a magnetic field.

The Altshuler-Lee-Stone theory of conductance fluc-tuations ceases to be applicable when the dimensionsof the sample approach the mean free path. In thisballistic regime observations of large aperiodic, as wellas quasi-periodic, magnetoconductance fluctuations havebeen reported.68,69,139,168,176,177,178,179 Quantum inter-ference effects in this regime are determined not by impu-rity scattering but by scattering off geometrical featuresof the device, as will be discussed in Section I.C.

D. Aharonov-Bohm effect

Magnetoconductance fluctuations in a channel geome-try in the diffusive regime are aperiodic, since the inter-fering Feynman paths enclose a continuous range of mag-netic flux values. A ring geometry, in contrast, enclosesa well-defined flux and thus imposes a fundamentalperiodicity

G() =G( + n(h/e)), n= 1, 2, 3, . . . , (2.48)

on the conductance as a function of perpendicular mag-netic field B (or flux = BS through a ring of areaS). Equation (2.48) expresses the fact that a flux in-crement of an integer number of flux quanta changes byan integer multiple of 2 the phase difference betweenFeynman paths along the two arms of the ring. The pe-riodicity (2.48) would be an exact consequence of gaugeinvariance if the magnetic field were nonzero only in theinterior of the ring, as in the original thought experimentof Aharonov and Bohm.180 In the present experiments,however, the magnetic field penetrates the arms of thering as well as its interior so that deviations from Eq.

FIG. 25 Illustration of the Aharonov- Bohm effect in a ringgeometry. Interfering trajectories responsible for the magne-toresistance oscillations with h/e periodicity in the enclosedflux are shown (a). (b) The pair of time-reversed trajecto-ries lead to oscillations with h/2e periodicity.

(2.48) can occur. Since in many situations such devi-ations are small, at least in a limited field range, onestill refers to the magnetoconductance oscillations as an

Aharonov-Bohm effect.The fundamental periodicity

B = h

e

1

S (2.49)

is caused by interference between trajectories that makeone half-revolution around the ring, as in Fig. 25a. Thefirst harmonic

B= h

2e

1

S (2.50)

results from interference after one revolution. A funda-

mental distinction between these two periodicities is thatthe phase of theh/e oscillations (2.49) is sample-specific,whereas the h/2e oscillations (2.50) contain a contribu-tion from time-reversed trajectories (as in Fig. 25b) thathas a minimum conductance at B = 0, and thus has asample-independent phase. Consequently, in a geometrywith many rings in series (or in parallel) the h/e oscilla-tions average out, but the h/2e oscillations remain. Theh/2e oscillations can be thought of as a periodic mod-ulation of the weak localization effect due to coherentbackscattering.

The first observation of the Aharonov-Bohm effect inthe solid state was made by Sharvin and Sharvin181 ina long metal cylinder. Since this is effectively a many-ring geometry, only the h/2e oscillations were observed,in agreement with a theoretical prediction by Altshuler,Aronov, and Spivak,182 which motivated the experiment.(We refer to Ref.125 for a simple estimate of the orderof magnitude of the h/2e oscillations in the dirty metalregime.) The effect was studied extensively by severalgroups.183,184,185 Theh/e oscillations were first observedin single metal rings by Webb et al.186 and studied theo-retically by several authors.1,144,187,188 The self-averagingof theh/e oscillations has been demonstrated explicitly in


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experiments with a varying number of rings in series.189

Many more experiments have been performed on one-and two-dimensional arrays and networks, as reviewed inRefs.190 and191.

In this connection, we mention that the developmentof the theory of aperiodicconductance fluctuations (dis-cussed in SectionII.C) has been much stimulated by theirobservation in metal rings by Webb et al.,165 in the course

of their search for the Aharonov-Bohm effect. The reasonthat aperiodic fluctuations are observed in rings (in ad-dition to periodic oscillations) is that the magnetic fieldpenetrates the width of the arms of the ring and is notconfined to its interior. By fabricating rings with a largeratio of radius r to width W, researchers have proven itis possible to separate190 the magnetic field scales of theperiodic and aperiodic oscillations (which are given by afield interval of order h/er2 and h/eWl, respectively).The penetration of the magnetic field in the arms of thering also leads to a broadening of the peak in the Fouriertransform at the e/h and 2e/h periodicities, associatedwith a distribution of enclosed flux. The width of the

Fourier peak can be used as a rough estimate for thewidth of the arms of the ring. In addition, the nonzerofield in the arms of the ring also leads to a damping ofthe amplitude of the ensemble-averagedh/2eoscillationswhen the flux through the arms is sufficiently large tosuppress weak localization.191

Two excellent reviews of the Aharonov-Bohm effect inmetal rings and cylinders exist.190,191 In the followingwe discuss the experiments in semiconductor nanostruc-tures in the weak-field regime c < 1, where the ef-fect of the Lorentz force on the trajectories can be ne-glected. The strong-field regime c > 1 (which is noteasily accessible in the usual polycrystalline metal rings)is only briefly mentioned; it is discussed more extensivelyin Section IV.D. To our knowledge, no observation ofAharonov-Bohm magnetoresistance oscillations in Si in-version layers has been reported. The first observation ofthe Aharonov-Bohm effect in a 2DEG ring was publishedby Timp et al.,69 who employed high-mobility GaAs-AlGaAs heterostructure material. Similar results wereobtained independently by Ford et al.73 and Ishibashi etal.193 More detailed studies soon followed.74,139,176,194,195

A characteristic feature of these experiments is the largeamplitude of theh/e oscillations (up to 10% of the aver-age resistance), much higher than in metal rings (wherethe effect is at best192,196,197 of order 0.1%). A similardifference in magnitude is found for the aperiodic mag-

netoresistance fluctuations in metals and semiconductornanostructures. The reason is simply that the amplitudeG of the periodic or aperiodic conductance oscillationshas a maximum value of order e2/h, so the maximumrelative resistance oscillation R/R RG Re2/h isproportional to the average resistance R, which is typi-cally much smaller in metal rings.

In most studies only the h/e fundamental periodic-ity is observed, although Ford et al.73,74 found a weakh/2e harmonic in the Fourier transform of the magne-

toresistance data of a very narrow ring. It is not quiteclear whether this harmonic is due to the Altshuler-Aronov-Spivak mechanism involving the constructive in-terference of two time-reversed trajectories182 or to therandom interference of two non-time-reversed Feynmanpaths winding around the entire ring.1,144,187 The rela-tive weakness of the h/2e effect in single 2DEG rings isalso typical for most experiments on single metal rings

(although the opposite was found to be true in the caseof aluminum rings by Chandrasekhar et al.,197 for rea-sons which are not understood). This is in contrast tothe case of arrays or cylinders, where, as we mentioned,theh/2e oscillations are predominant the h/e effect be-ing ensemble-averaged to zero because of its sample-specific phase. In view of the fact that the experimentson 2DEG rings explore the borderline between diffusiveand ballistic transport, they are rather difficult to ana-lyze quantitatively. A theoretical study of the Aharonov-Bohm effect in the purely ballistic transport regime wasperformed by Datta and Bandyopadhyay,198 in relationto an experimental observation of the effect in a double-quantum-well device.199 A related study was publishedby Barker.200

The Aharonov-Bohm oscillations in the magnetoresis-tance of a small ring in a high-mobility 2DEG are quiteimpressive. As an illustration, we reproduce in Fig. 26the results obtained by Timp et al.201 Low-frequencymodulations were filtered out, so that the rapid oscil-lations are superimposed on a constant background. Theamplitude of theh/eoscillations diminishes with increas-ing magnetic field until eventually the Aharonov-Bohmeffect is completely suppressed. The reduction in am-plitude is accompanied by a reduction in frequency. Asimilar observation was made by Ford et al.74 In metals,in contrast, the Aharonov-Bohm oscillations persist tothe highest experimental fields, with constant frequency.The different behavior in a 2DEG is a consequence ofthe effect of the Lorentz force on the electrons in thering, which is of importance when the cyclotron diame-ter 2lcycl becomes smaller than the width Wof the armof the ring, provided W < l (note that lcycl = hkF/eBis much smaller in a 2DEG than in a metal, at the samemagnetic field value). We will return to these effects inSectionIV.D.

An electrostatic potential V affects the phase of theelectron wave function through the term (e/h)

V dt in

much the same way as a vector potential does. If thetwo arms of the ring have a potential difference V, andan electron traverses an arm in a time t, then the ac-quired phase shift would lead to oscillations in the re-sistance with periodicity V = h/et. The electrostaticAharonov-Bohm effect has a periodicity that depends onthe transit time t, and is not a geometrical property ofthe ring, as it is for the magnetic effect. A distributionof transit times could easily average out the oscillations.Note that the potential difference effectuates the phasedifference by changing the wavelength of the electrons(via a change in their kinetic energy), which also distin-


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FIG. 26 Experimental magnetoresistance of a ring of 2 m di-ameter, defined in the 2DEG of a high-mobility GaAs-AlGaAsheterostructure (T = 270 mK). The different traces are con-secutive parts of a magnetoresistance measurement from 0to 1.4 T, digitally filtered to suppress a slowly varying back-ground. The oscillations are seen to persist for fields wherec > 1, but their amplitude is reduced substantially formagnetic fields where 2lcycl W. (The field value where2lcycl

2rc = W is indicated). Taken from G. Timp et al.,

Surf. Sci. 196, 68 (1988).

guishes the electrostatic from the magnetic effect (wherea phase shift is induced by the vector potential withouta change in wavelength). An experimental search for theelectrostatic Aharonov-Bohm effect in a small metal ringwas performed by Washburn et al.202 An electric fieldwas applied in the plane of the ring by small capacitiveelectrodes. They were able to shift the phase of the mag-

netoresistance oscillations by varying the field, but theeffect was not sufficiently strong to allow the observa-tion of purely electrostatic oscillations. Unfortunately,this experiment could not discriminate between the ef-fect of the electric field penetrating in the arms of thering (which could induce a phase shift by changing thetrajectories) and that of the electrostatic potential. Ex-periments have been reported by De Vegvar et al.203 on

the manipulation of the phase of the electrons by meansof the voltage on a gate electrode positioned across oneof the arms of a heterostructure ring. In this system achange in gate voltage has a large effect on the resistanceof the ring, primarily because it strongy affects the localdensity of the electron gas. No clear periodic signal, in-dicative of an electrostatic Aharonov-Bohm effect, couldbe resolved. As discussed in Ref.203, this is not too sur-prising, in view of the fact that in that device 1D subbanddepopulation in the region under the gate occurs on thesame gate voltage scale as the expected Aharonov-Bohmeffect. The observation of an electrostatic Aharonov-Bohm effect thus remains an experimental challenge. Asuccessful experiment would appear to require a ring inwhich only a single 1D subband is occupied, to ensure aunique transit time.198,200

E. Electron-electron interactions

1. Theory

In addition to the weak localization correction tothe conductivity discussed in Section II.B, which arisesfrom a single-electron quantum interference effect, theCoulomb interaction of the conduction electrons gives

also rise to a quantum correction.

204,205

In two dimen-sions the latter correction has a logarithmic tempera-ture dependence, just as for weak localization [see Eq.(2.14)]. A perpendicular magnetic field can be usedto distinguish the two quantum corrections, which havea different field dependence.118,204,205,206,207,208,209,210

This field of research has been reviewed in detail byAltshuler and Aronov,211 by Fukuyama,212 and by Leeand Ramakrishnan,127 with an emphasis on the theory.A broader review of electronic correlation effects in 2Dsystems has been given by Isihara in this series.213 In thepresent subsection we summarize the relevant theory, asa preparation for the following subsection on experimen-tal studies in semiconductor nanostructures. We do not

discuss the diagrammatic perturbation theory, since it ishighly technical and does not lend itself to a discussionat the same level as for the other subjects dealt with inthis review.

An attempt at an intuitive interpretation of the Feyn-man diagrams was made by Bergmann.214 It is arguedthat one important class of diagrams may be interpretedas diffraction of one electron by the oscillations in theelectrostatic potential generated by the other electrons.The Coulomb interaction between the electrons thus in-


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the negative conductivity correctionee. This effect caneasily be studied up to c= 10, which would imply anenhancement by a factor of 100 of the resistivity correc-tion in zero magnetic field. (The Hall resistivity xy alsocontains corrections fromee, but without the enhance-ment factor.) In 2D it is this enhancement that allowsthe small effect of electron-electron interactions to be ob-servable experimentally (in as far as the effect is due to

diffuson-type contributions).Experimentally, the parabolic negative magnetoresis-

tance associated with electron-electron interactions wasfirst identified by Paalanen et al.137 in high-mobilityGaAs-AlGaAs heterostructure channels. A more detailedstudy was made by Choi et al.55 In that paper, as wellas in Ref.113, it was found that the parabolic magnetore-sistance was less pronounced in narrow channels thanin wider ones. Choi et al. attributed this suppressionto specular boundary scattering. It should be noted,however, that specular boundary scattering has no ef-fect at all on the classical conductivity tensor 0 (inthe scattering time approximation; cf. Section II.A.2).

Since the parabolic magnetoresistance results from the(c)2 term in 1/0xx [see Eq. (2.54)], one would expect

that specular boundary scattering does not suppress theparabolic magnetoresistance (assuming that the resultxy = yx = 0 still holds in the pure metal regimel > W). Diffuse boundary scattering does affect0, butonly for relatively weak fields such that 2lcycl > W (seeSectionII.A); hence, diffuse boundary scattering seemsequally inadequate in explaining the observations. In theabsence of a theory for electron-electron interaction ef-fects in the pure metal regime, this issue remains unset-tled.

2. Narrow-channel experiments

Wheeler et al.38 were the first to use magnetoresis-tance experiments as a tool to distinguish weak localiza-tion from electron-electro

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