VOLUME 63, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 1989

Quenching of the Hall Resistance in Ballistic Microstructures: A Collimation Effect

Harold U. BarangerATdk T Bell Laboratories, 4G-314, Holmdel, New Jersey 07733

A. Douglas StoneSection of Applied Physics, Yale University, Box 2I57, New Haven, Connecticut 06520

(Received 16 February 1989)

We present both calculations and a physical interpretation of the observed suppression (quenching) ofthe low-field Hall resistance in quasi-one-dimensional ballistic microstructures. We find that quenchingis due to a property of the contact geometry and is not intrinsic to the quasi-one-dimensional limit. Gen-eric quenching, as observed experimentally, is found only when the width of the wires is gradually in-creased near the junction to the Hall probes. The resulting collimation of the electrons in the forwarddirection reduces the sensitivity to a magnetic field.

PACS numbers: 72.20.My, 73.20.Dx, 73.40.Kp

Several recent transport experiments have demonstrat-ed novel quantum-mechanical eA'ects in small pure sam-

ples where the transport is ballistic. Examples includethe nonlocal "bend resistance, "' electron-focusing finestructure, and anomalous quantization of the Hall resis-tance in four-probe measurements and the quantizedpoint-contact resistance in two-probe measurements, allof which have received adequate theoretical explana-tion. ' ' However, one of the earliest of thesediscoveries, the disappearance or "quenching" of thelow-magnetic-field Hall resistance, has resisted a com-plete theoretical explanation.

Roukes et al. observed that the Hall resistance, R~,of very narrow high-mobility wires (width =100 nm)measured in the standard cross geometry typically is

suppressed below the expected two-dimensional valuenear zero field. Later work confirmed the eA'ect andclearly indicated that it was generic in the sense that itoccurred for diff'erently fabricated ballistic microstruc-tures over a wide range of carrier densities. ' '' Typi-cally, Rz versus magnetic field, although not completelystructureless, has an average slope near zero when 8 isless than a characteristic value which varies with thewidth of the wire. Earlier theoretical work' argued thatthe eAect should always occur in the quasi-one-dimensional limit since there is a crossover length scale,3kth =3(h/2kFeB) ' ', at which the electronic states atthe Fermi energy change from being extended across thewire to being localized by the Lorentz force at one edge.It was proposed but not demonstrated that a wire withwidth less than this length scale should always showquenching. Subsequent microscopic calculations assum-ing idealized weakly coupled Hall probes' ' failed toshow quenching. Our calculations below show thatquenching is not an intrinsic property of the Hall resis-tance of narrow wires even with strongly coupled probes.

We find that generic quenching only arises due to aproperty of the contact geometry, namely, the wideningof the wires near the junction region. This widening,

which is typically present in quasi-one-dimensional mi-crostructures but need not be, acts like an acoustic hornand sets up a nonequilibrium distribution in which modeswith high longitudinal momentum are preferentially pop-ulated. The resulting collimation of the electrons inject-ed into the junction suppresses RH since, as shownbelow, electrons in these modes contribute little to RH.

The first step in calculating the resistance of a particu-lar phase-coherent structure in a magnetic field is to finda correct expression for the four-probe resistance, whichis valid before spatial or impurity averaging, and is ableto account for a specific measuring geometry. Otherwork' ' ' ' has relied on the Landauer-type formula-tion proposed by Buttiker' which, although physicallyappealing on a number of grounds, had not been con-nected to standard linear-response theory. The Buttikerapproach considers the linear response of a multiprobesystem to fixed potentials V„applied to leads n whichcause current to flow in leads m according toI„,=gg „V„. His fundamental result is that g ~ =T „(at T=O for m&n), where T „ is the sum of all thetransmission coef5cients from lead n to m between themodes at the Fermi surface. Since in the presence of amagnetic field there are always circulating currents aris-ing from states below the Fermi surface, it was not obvi-ous that this expression was correct in arbitrary magnet-ic field.

We have recently shown that Buttiker's result ' can bederived rigorously from linear-response theory in an ar-bitrary magnetic field; details will be presented else-where. ' Our calculation leads to expressions for thenonlocal conductivity tensor, a(x,x'), and for the g „ in

terms of the exact eigenstates or Green's functions of thenoninteracting problem. We find that although a(x,x')(which describes both diamagnetic and transportcurrents) does depend on all filled states, the g „(andhence RH) are determined solely by wave functions atthe Fermi surface, and can be transformed to yield pre-cisely Buttiker's result. For a symmetric cross structure

414 1989 The American Physical Society

VOLUME 63, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 1989

in which each lead supports N, h,. „ transverse channels atthe Fermi energy, the Hall resistance is

TR TL

e' 2TF(TF +T'R+ TL)+ TR+ TL

(a)

Wwhere TL, TR, and TF are the transmission intensities toturn left, turn right, and go forward, respectively, at theFermi energy. We find the transmission intensities need-ed in Eq. (1) by applying the recursive Green's-functionmethod to a nearest-neighbor tight-binding Hamiltonianfor a square lattice of the desired geometry. '

The first structure that we study is a cross formed ofperfect straight wires of width 8'with infinite hard walls(Fig. 1). Very recently, other groups ' ' have present-ed results for similar structures; we present our indepen-dent results here since the comparison of the straightwire case and the graded-width wire case is essential fora clear understanding of quenching.

The Hall-resistance traces in Fig. 1(a) show that avariety of behavior occurs in the perfect cross, includingquenching, a slope greater than the two-dimensionalvalue, and at higher fields the quantum Hall effect.(Quantum Hall plateaus for the multichannel casesshown are off scale. ) To present the behavior in the low-field regime more systematically, Fig. 1(b) shows theslope of the RH(B) curve near B=O normalized to thetwo-dimensional value. There is a great deal of structurein this curve including cusps at the thresholds of the sub-bands; ' the energy range shown is from threshold up toN, h„. „=6. It is apparent that quenching of RH occursonly in very restricted ranges of energy for this structureand only for N, h„. „~2. For higher channel fillings, RHessentially oscillates around its classical value andreasonable averages over energy to include the eA'ect oftemperature give approximately the classical value. Thesame qualitative features are found for junctions withharmonic instead of hard-wall confinement. ' ' Thusthe behavior of these structures, and hence the mecha-nism discussed in Ref. 16, does not explain the genericquenching observed experimentally '' which in almostall cases corresponds to the range N, h,. „~3. Further-more, Fig. 1(b) demonstrates that the slope found usinga very small magnetic field (3k,h) 100M, solid) agreeswith that fit over a broad range of fields (up to3k,h= W/2, dotted). Since the broad field range in-cludes values well above the quenching threshold of Ref.12 while the small field is far below this threshold, thisdemonstrates that the length-scale arguments proposedpreviously ' are not relevant to the occurrence ofquenching.

To obtain generic quenching it is crucial to include thewidening of the wires near the junction. While the elec-trostatic potential profile for a junction of two such wireshas not been calculated or measured, it is certain thatthere is considerable rounding of the corners at the junc-tion. We include this in our structures by slowly increas-ing the width of each lead as the junction is approached

m 0X

CC

C3Z'.

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0CC

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1 I I I » I

1 4 7

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0(b)

—1.0 —0.5I

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I

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1.5

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F1G. l. (a) The product of the Hall resistance and the den-

sity as a function of the magnetic field for the perfect hard-wall cross structure shown. At different energies, RH quenchesbefore rising to the n= 1 quantum Hall plateau (EF =2.7E~,N, h,„=1, solid line) is greater than the two-dimensional value(EF 14E~, N, h„=3, dotted line) or is close to the 2D value

(EF =20E ~, N,h„=4, dash-dotted line). The 2D value is

dashed and El is the threshold energy of the lowest transversesubband. Using T summed over the lowest N' channels, theinset shows RH(N') for a N, h, , =8 case (EF =20E~ with twicethe width). (b) The slope of the Hall resistance S normalizedto its 2D value as a function of Fermi energy. Quenchingoccurs only in narrow regions. The slope at low field (solidline, BW /&0=4x10 ) agrees with that obtained by fittingRH(B) over a broad range of fields [ —8~,8~] (dotted line,8 ~ W /d&p =0.6 for EF & 14E ~, 8 ~ W /&Po =0.8 for EF ) 14E ~).

[Fig. 2(a) j.'oThe Hall-resistance trace in Fig. 2(a) shows that

strong quenching is possible in the graded-width struc-ture with N, h„, „=4. The T=O trace shows a great deal

415

VOLUME 63, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 1989

0.15

0.10

-=--:-&11/111111

0.05

0)

X

CC

~ ~

—0.05

—0.10

—0.15—1.5

I

0.0I

0.5I

1.0

(a)

1.5

1.5

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0.5—

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—0.5—

—1.00I

10 20EF/E1

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30

(b)

40

FIG. 2. (a) The Hall resistance as a function of magneticfield f'or a graded-width structure at Ep =20E]. RH showsstrong quenching (T=O, dotted line; T=3E~, solid line; two-dimensional value, dashed line). The structure consists of ahard-wall narrow wire which gradually widens (hatched) to awire of twice the original width with some softness in the sidewalls in the junction region (shaded). (b) The slope of theHall resistance normalized to its two-dimensional value as afunction of Fermi energy. Quenching occurs over a broadrange of energies (T=O, dotted line; T=E|, solid line). Slopesare obtained by fitting RH(B) as in Fig. 1(b).

of fine structure as a function of magnetic field eventhough there is no disorder. Such fine structure is ob-served experimentally as the temperature is lowered andhad previously been ascribed to impurity effects. ' Ourcalculation indicates that such structure may be intrinsicand distinct from diffusive conductance fluctuations.

The crucial result, that quenching is a generic featureof the graded-width structure, is evident from the nor-malized slopes in Fig. 2(b). While there is still substan-

416

tial modulation of the slope with Fermi energy at T=O,the temperature average is close to zero and certainly farbelow the two-dimensional value. We have studied manydifferent graded-width structures, e.g. , both hard-walland harmonic transverse confinement, with or without asoftening of the potential in the junction region, and withdifferent types of grading as a function of length alongthe wire. All show generic quenching, in some caseseven if the smooth grading is only over a distance of or-der of the junction size (a realistic scale for experimentalsystems). However, some region of gradual widening isessential in producing this quenching behavior; an abruptchange from a narrow to a wide wire produces a normal-ized slope which modulates about l.

These results suggest a natural physical explanationfor the quenching of R&. A gradual widening of thewires causes injected electrons to travel adiabaticallyfrom the narrow to the wide region so that their trans-verse quantum state (channel number) is conserved.This leads to a transfer of transverse momentum k& intolongitudinal momentum kII. Thus the current that wasinjected into the narrow wire equally distributed acrossall channels (i.e., consistent with a fixed chemical poten-tial in the reservoir) reaches the wide-junction regiondistributed across only the low-lying channels; these arethe channels with large k~~/k&. Ultimately the wideningof the equipotential lines at the junction must becomenonadiabatic, since they must turn through 90 . Fromthis point on, one expects k& to be conserved instead ofthe mode number, as demonstrated for an abruptly ter-minated constriction, yielding a collimated beam ofelectrons in the forward direction, as discussed recentlyfor a constriction.

What is the effect of this collimation on RH? It isuseful to rewrite Eq. (1) as RH = a,TRL/D, wherea = (T~ —TL )/(Tg+ TL ) is the fractional right-lefttransmission asymmetry, T~I = T&+ TI is the total side-ways transmission, and

D =(e /h) [2TF(TF+ T~L)+ Tgl (1+tr')/2]

is relatively insensitive to whether the electrons aretransmitted forwards or sideways. It is then clear thatRH can be suppressed either by (1) reducing the frac-tional asymmetry a due to the field, or simply by (2) in-creasing the forward transmission at the expense of side-ways transmission (since RH 0 as TRL 0 at fixed a).Collimation will obviously produce the latter effect, atleast up to a threshold field at which the collimated beamis bent into a voltage probe, and we find numerically thatgrading also reduces e. To analyze the effect of gradingon RH, we define two quantities which measure the im-portance of each mechanism separately: Ri =a&(T&L/D)s is the asymmetry from the graded-wire calculationmultiplied by the magnitude factor from the straightwire case, while R2= as(TRL/D)G incorporates theeffect of grading on the magnitudes but uses c forstraight wires. Calculating the normalized slopes of

VOLUME 63, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 1989

these approximate Hall resistances exactly as for thetrue RH and averaging over EF for which there are 3-5modes present, we find that (S~/S2o) =0.45 ~0.16 and(Sz/S2o) =0.29~0.02, while for the true &H, (Ss«d, d/

S2o) =0.18+ 0.05 and (S„,„. ;sh, /S2o) =0.92+ 0.02.Thus we see that grading causes quenching through bothmechanisms noted above.

In fact, if we analyze in more detail the transmissionintensities for the uniform cross (which does not showgeneric quenching), we can make a plausibility argumentthat an adiabatically graded cross of the same finalwidth should show generic quenching (as in Fig. 2). Ifthe uniform cross had been graded from an initial widthwith only N' propagating channels, then, roughly speak-ing, only the lowest N' channels would be occupied at thejunction, and to estimate the change in the Hall resis-tance which would result we use the same transmissionintensities as for the uniform case, but only sum over thelowest N' incoming and outgoing channels. In Fig. 1(a)we plot RH as a function of N' (N, h,, „=8);if we are try-ing to estimate R~ for a graded cross which doubles inwidth (as in Fig. 2), N'=4 is the appropriate value toconsider, and for this value RH is indeed stronglyquenched. The fact that RH(N') for the uniform crossjumps up sharply at N' ~ 6 directly verifies that the hightransverse channels make the dominant contribution toR~.

In conclusion, generic quenching is present in struc-tures when there is an approximately adiabatic gradingfrom a narrow wire to a larger junction region in orderto collimate the electrons. An obvious experimental im-plication of our explanation is that the geometry of thejunction should be crucial in determining whetherquenching is present or not. Very recently, the depen-dence of quenching on the geometry of the junction hasbeen studied experimentally by Chang and Chang andFord et ah. ' Change and Chang showed that a uniformnarrow wire is not necessary to produce generic quench-ing; instead a constriction on each lead of a wide crossstructure is a necessary and sufhcient condition forstrong quenching, a result consistent with our modelsince the constriction introduces the necessary narrow re-gion before the wide junction. In one geometry studiedby Ford et al. , a deflector is placed in the center of thejunction. This sample showed no quenching at all, asone would expect from our model since the deflectorprevents forward transmission of the collimated beam.In another geometry in which the lithographic junction isdiamond shaped, they found the sign of RH inverted; ourcalculations for this geometry indicate that inversiononly occurs if there is collimation. Together all these ex-perimental and theoretical results indicate conclusivelythat junction geometry, and specifically the collimationeAect, is the basic mechanism for the observed genericquenching of the Hall resistance. The creation of anonequilibrium momentum distribution by the samplegeometry appealed to in our explanation of the quench-

ing appears to be a concept which unifies many of thenovel quantum phenomena' observed in ballistic trans-port.

%'e acknowledge valuable discussions with M.Biittiker, A. M. Chang, M. L. Roukes, A. Szafer, and G.Timp. Y. Imry was helpful in clarifying the crucial roleof the abrupt termination in causing collimation. Thework at Yale was supported in part by NSF Grant No.DMR-8658135 and by the ATILT Foundation.

'G. Timp et al. , Phys. Rev. Lett. 60, 2081 (1988).~H. van Houten et a!., Europhys. Lett. 5, 721 (1988).3M. Biittiker, Phys. Rev. B 38, 9375 (1988); B. J. van Wees

et al. , Phys. Rev. Lett. 62, 1181 (1989).~B. J. van Wees et al. , Phys. Rev. Lett. 60, 848 (1988); D.

A. Wharam et al , J. Phys. C. 21, L209 (1988).5C. W. J. Beenakker, H. van Houten, and B. J. van Wees,

Europhys. Lett. 7, 359 (1988).6L. I. Glazman et al. , Pis'ma Zh. Teor. Fiz. 48, 218 (1988)

[JETP Lett. 48, 238 (1988)];G. Kirczenow, Solid State Com-mun. 68, 715 (1988); A. Szafer and A. D. Stone, Phys. Rev.Lett. 62, 300 (1989); E. G. Haanappel and D. van der Marel,Phys. Rev. B 39, 5484 (1989).

M. L. Roukes, A. Scherer, S. J. Allen, Jr. , H. G. Craighead,R. M. Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev.Lett. 59, 3011 (1987).

~C. J. B. Ford, T. J. Thornton, R. Newbury, M. Pepper, H.Ahmed, D. C. Peacock, D. A. Ritchie, J. E. F. Frost, and G. A.C. Jones, Phys. Rev. B 38, 8518 (1988).

9A. M. Chang and T. Y. Chang (to be published).' C. J. B. Ford, S. Washburn, M. Buttiker, C. M. Knoedler,

and J. M. Hong, Phys. Rev. Lett. 62, 2724 (1989).' 'M. L. Roukes (private communication).'2C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett.

60, 2406 (1988).'3F. M. Peeters, Phys. Rev. Lett. 61, 589 (1988).'~H. Akera and T. Ando, Phys. Rev. B 39, 5508 (1989).'~D. G. Ravenhall, H. W. Wyld, and R. L. Schult, Phys. Rev.

Lett. 62, 1780 (1989).'6G. Kirczenow, Phys. Rev. Lett. 62, 2993 (1989).'7M. Biittiker, Phys. Rev. Lett. 57, 1761 (1986).'"H. U. Baranger and A. D. Stone (to be published).'9P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981);

H. U. Baranger, A. D. Stone, and D. P. DiVincenzo, Phys.Rev. B 37, 6521 (1988).

The width in the straight wire case is 21 sites; the graded-width wire varies from 21 to 41. To grade, we introduce alinear potential in the shaded region in the inset to Fig. 2(a)whose slope starts large and becomes small near the junction;the slope in the junction region is El per site for Fig. 2. The 8field is zero far from the junction and graded to the desiredvalue within W of the junction; the nonuniformity of 8 does notaftect the low-field properties.

'H. U. Baranger and A. D. Stone, in "Science and Engineer-ing of 1- and 0-Dimensional Semiconductors, " edited by S. P.Beaumont and C. M. Sotomayor-Torres (Plenum, New York,to be published).

22Szafer and Stone, Ref. 6.C. W. J. Beenakker and H. van Houten, Phys. Rev. B 39,

10445 (1989).

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