Propagation around a Bend in a Multichannel Electron Waveguide

VOLUME 60, NUMBER 20 PHYSICAL REVIEW LETTERS 16 MAY 1988

Propagation around a Bend in a Multichannel Electron Waveguide

G. Timp, H. U. Baranger, P. deVegvar, J. E. Cunningham, R. E. Howard, R. Behringer,and P. M. Mankiewich

ATd'c T Bell Laboratories, Holmdel, New Jersey 07733(Received 16 December 1987)

We have measured the four-terminal magnetoresistance of an electron waveguide, i.e., a high-mobilitywire fabricated in GaAs-A16aAs heterostructure in which only a few transverse one-dimensional sub-

bands carry the current, and simulated the measurement numerically. We observe that the averageresistance increases when the current path bends through a junction at or beyond the voltage terminalsbecause of a change (due to the bend) in the relative transmittance of the few subbands. Our observa-

tions reveal that the average resistance is predominantly associated with scattering from the leads used

for the measurement.

PACS numbers: 72. 15.6d, 73.60.Br

The interpretation of a four-terminal magnetoresis-tance measurement performed on a submicron scale atlow temperature is problematic because the leads are in-

vasive. In this regime the resistance of a conductor is

directly related to the quantum mechanical transmissionof carriers. ' If a carrier propagates into a lead while

maintaining phase coherence, then the lead must be in-

cluded in the computation of the resistance. While theview that the leads are an integral part of the four-terminal resistance is now widely appreciated, it is theeff'ect of impurity scattering in the leads on fluctuationsin the magnetoresistance which has been the focus of at-tention, rather than the eff'ect of the lead on the averageresistance. Generally, a lead elastically scatters an elec-tron as an impurity would, but a preponderence of im-

purity scattering and transverse subbands usually ob-scures the effect of a lead.

In this Letter, we describe our four-terminal measure-ments of the magnetoresistance of an electron wave-

guide, i.e., a quasi one-dimensional (1D) wire which is

only a few Fermi wavelengths wide and comparable in

length to the elastic mean free path estimated from themobility. We observe that the average, symmetrizedmagnetoresistance consistently increases when the cur-rent path bends through a junction either at the voltageterminals or beyond the terminals in correspondencewith our numerical simulation of the measurement. Thisincrease in the average four-terminal resistance is attrib-uted to a change in the relative transmittance throughthe bend of the few current-carrying channels ortransfers subbands, and to the lack of scattering by im-

purities. Our observations reveal that in these devicesresistance is predominantly due to scattering from theleads used for the measurement.

Since the fabrication of the 1D wires has already beendescribed elsewhere, we will discuss only the salientfeatures here. The devices were fabricated in both6-doped and conventional selectively doped GaAs-Alp 3pGaQ 7QAs heterostructures prepared by molecular-

beam epitaxy. A typical heterostructure comprises a7-nm GaAs cap layer, an undoped AIGaAs layer 20-40nm thick, a doped layer, and an A1GaAs spacer layer d,thick grown on top of GaAs. The mobility, p, and car-rier density, n20, were determined at 280 mK with aHall bridge geometry 300 pm wide and 2 mm long.Electron-beam lithography is used to pattern an etchmask on the heterostructure. This protects the underly-ing doped A1GaAs from a partial etch which laterallyconfines the electron gas to the region beneath the mask.Although the lithographic linewidth of the mask is 500nm, the conduction width of the wire is only about100-200 nm because of edge depletion. We have not ob-served any degradation of the mobility with this fabrica-tion technique.

We measured the four-terminal, transverse (to thefield) magnetoresistance of devices like the one shownschematically in Fig. 1(a) using an ac resistance bridgeoperating at 14 Hz. If the same lead configuration wasused, the measured resistance was correlated from day today to better than 99.6%, even after the leads had beenchanged several times. The resistance measured at H 0for T =4 K did not generally scale with the separationbetween voltage terminals, e.g. , it depends on bends in

the current path (see below), but the high-field (H ) 2T) quantized Hall effect measured at every junction in

the pattern and the corresponding Shubnikov-de Haasoscillations observed in every segment [see Fig. 1(a)]were characteristic of a homogeneous device.

The conducting width of the wire, 8; and the inelasticdiffusion length, L&, were estimated from the magne-toresistance of nominally 2-pm-diam annuli with alinewidth of 500 nm fabricated from the same materialsin the same way. Peaks in the Fourier spectra of themagnetoresistance of the annuli are observed at approxi-mately 720 and 1450 T ' corresponding to fluxes ofhc/e and hc/2e penetrating the area of the annulus.The width of the hc/e fundamental in the spectra is as-sumed to be given approximately by hH ' =2rrrW(hc/

1988 The American Physical Society 2081


1.0

1 6 Ni 20Pl

,o8 iv 1 )( ioP)

t Biked i

0 6 MS(ilp~i r V~, l i

0 . .4 8 12 I & IA

H(T) ~ (

. ,~' l 1'1~ -,h, l

ht Ml 1 t~ ., 1

r~i& I ..j"'i l, ~&

' ~l'

02 Wl t'

-400 -20061 5

0H (mT)

(a)

200

0.6

1.2

0.8

0.4

—0.4 cc

1,2

—0.8L4

—1.2400

RS0 IIfll

—200 0 200Rs H(mT)

0.4—

tx3 0.2

CC

RA

—0.2-400 4000 200

H (mT)

(b)FIG. 1(a). The resistances R3968 and R39,8 io measured as

functions of magnetic field at 280 mK in device No. 1. Lower

right inset: The cross pattern of a typical device. The center-to-center separations between probes in the asyrnrnetric crosspattern are (clockwise from 12) 2.2@m, 900 nm, 2.5 pm, 700nm. Upper left inset: The magnetoresistance over a largerrange in magnetic field. (b) The symmetric and antisymmetriccomponents of R39 68. Inset: The same decomposition ofR 39,S-IO.

the voltage terminals in the inset of Fig. 1(a)] is grosslyasymmetric, and R39g-lp is nonzero at H=O. For themeasurement denoted R396g the voltage terminals onthe device are separated by 700 nm, and the current isinjected along a straight line through the junction be-tween the voltage and current leads and exits straightthrough the second junction. The zero-field position wasdetermined from a separate Hall probe in close proximi-

ty with the device, but it should be apparent that notranslation of the field axis could result in R396g(H)=R396g( H). Moreover, we observe that the resis-tance is negative near H = 120 m T.

When the current and voltage leads are interchanged,we find that Rkl „(H)=R „kl( H) —approximately,which suggests the decomposition' R =

2 [Rkl, (H)+R n, kl(H)] alld R =

2 [Rkl, mn(H) R n, kl(H)].The results of the decomposition of R39 6g are shown in

Fig. 1(b). R is approximately (within 10%) symmetricabout H=O; R" is approximately antisymmetric; andthere is no negative resistance in Rf9sg. If we decom-pose R39 g lp we find [see the inset to Fig. 1(b)] that Rcan be negative, while R" is linear in field beyond 500mT with slope corresponding to a carrier density of2.4X 10" cm . The entry n; =2 in Table I refers to thisslope. Generally, below 100 mT the Hall effect is

suppressed.Thus, the magnetoresistance is composed of fluctuat-

ing symmetric and asymmetric components. ' " Therms amplitude of the asymmetric fluctuations, 8R"(BR"= 70, 390, and 35 0 for samples No. 1, No. 2, andNo. 3 respectively), can be comparable in magnitude toR~, as well as the rms amplitude of the symmetric fluc-tuations, BR~. Both BR~ and BR are approximately in-

dependent of length for 2.5 pm & L & 700 nm. 2

The asymmetry in the magnetoresistance, the negativeresistance, and the finite Hall effect at H =0 observed in

our four-terminal measurements do not violate Qnsager'ssymmetry relations because the resistance is actually an-isotropic (it depends on the leads and impurity configur-ation) as well as nonlocal for Ll, & L where L is the dis-

e) ' where r is the radius, and W is (a lower bound on)the conducting width of the wire. Once W is estimatedfrom /3H ', a lower bound on the number of transversesubbands, NT, is calculated with use of a hard confining

potential. The number of obvious Hall plateaus found in

a measurement such as R39 g-lp (see below) is consistentwith this estimate for NT. L~ is estimated from the ratioof the peak Fourier amplitudes associated with hc/e and

hc/2e obtained for H & 20 mT with the assumption thatthe ratio is proportional to exp(2rrr/L~).

Figure 1(a) shows an example of the magnetoresis-tance observed about H =0. The resistance R39 6g

[where 3 and 9 refer to the current terminals, 6 and 8

Sample identification number1 2 3

n2o/10" (cm ')p/10 (cm /V s)n, =2/10'' (cm )d, (nm)W (nm)Nr

(um)L, (nm)Rp(n)

3.3 ~0.1

18+ 1

2.4 + 0. 1

36140-150

4-5& 3.2

420 W 140670+ 100

4.2 + 0.1

8.7 + 0.22.0 + 0.1

27100-110

2-3& 3.7

360 w 902700 + 550

3.9 ~ 0.1

3.0 + 0.22.6 + 0.1

14120-140

4-5& 4.7

700+ 50

TABLE I. A list of the measured parameters (referred to in

the text) for the three materials examined.

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2000—

1600—

1200—

800

10

'«&I J,~300 y ~~ y

R"

I i=212 I ~I E (4{~i%Et'~V" ilvg'II'~r I

Il -300 3000

H(m T)10 20

3L (pm)3

4k~. . ii=31.0. . .II, g

-400 -200

(a)

I,j =3,9

0

H (mT)

200 400

I l I

llll

\

l\

l\

1 0

0I

2 3

MODE INJECTED

2 3 6 7 8

tu

(b)FIG. 2. (a)R found at 280 mK with use of different

current terminals for the same voltage terminals separated by900 nm in device No. 1. Upper right inset: The symmetricand asymmetric decompositions of R3.|Q,46 (solid line) and

R3s46 (dashed line). Upper left inset: The decay of AR vs

the distance between the bend in the current probe and thenearest voltage terminal for device No. l. (b) The calculatedchange in the average resistance vs number of modes (seetext). Inset: The normalized current transmitted out each ofthe two exit ports of a "T" (straight through, solid line;around a corner, dashed line) when a specific mode is injecteddown the main channel.

tance between voltage probes, so that the usual four-terminal resistance is not trivially related to thecoefficients to which the symmetry relations apply. Theresistance is invariant under a field reversal when accom-panied by the exchange of the current and voltage termi-nals because of the microscopic reciprocity of thetransmission coefficients. ' This invariance has alreadybeen demonstrated~ in the four-terminal magnetoresis-

tance of normal metal wires and loops where L~& L& 8'»L, with NT = 10 .

We observe that the average, symmetric magnetoresis-tance changes dramatically if the current path bendsthrough a junction at the voltage terminals or beyond theterminals. Figure 2(a) shows the changes in the four-terminal, symmetric magnetoresistance observed forvoltage terminals 900 nm apart when the current pathbends through a junction 700 nm from the voltage termi-nals (Rs lo 64), at one voltage terminal (Ri 12 64), and atboth voltage terminals (R2 12 64) relative to a geometryin which the current is injected through a junction alonga straight line (R3964). The four-terminal resistance is

always observed to increase at zero field (in the 52 casesexamined) whenever the current path bends at a voltageprobe, if the mobility of the starting heterostructureexceeds p =3x10 cm /V s with 8'=100 nm and

NT = 5, whereas no change is observed outside the am-

plitude of fiuctuation when @=4.7xi0 cm /V s withW= 180 nm corresponding to NT = 8-10. Further-more, the resistance changes for p ) 8.7x 10 cm /V s

and NT = 2-5 when the current bends through a junc-tion separated from the voltage terminals by at least 700nm, i.e., there is nonlocal effect on the average resis-tance. The change in resistance with magnetic field con-sistently decays within about 200 mT for each of the de-

vices examined. At 200 mT, r, =(eH/hc) 'i = W/2 andthe magnetic potential dominates over the electrostaticconfinement.

The change in resistance due to a bend in the currentpath, AR~, observed as a function of the distance be-tween the nearest voltage terminal and the junctionwhere the current path bends, AL, is plotted for deviceNo. 1 in the left inset of Fig. 2(a). The change is con-sistent with the law BR~ =Roexp( dL/L, ) wh—ere Ro is

the change in resistance measured when the currentbends through a junction at one voltage terminal, and L,is a characteristic length on the order of 500 nm (seeTable I). Generally, the same value of Ro is obtainedthroughout the device independent of the leads used tomeasure it, within the amplitude of the fiuctuations. (Inaddition, the value of Ro obtained in the annuli used toestimate the two parameters, W and L~, are the same asthe values obtained throughout the asymmetric crosspattern within the Auctuations. )

We attribute our observations to differences in thetransmission coefficients for propagation straight througha junction and around a bend in a junction. To justifythis we calculate the transmission coefficients at the Fer-mi energy in a model multiprobe microstructure wherethe electrons are confined by hard walls. ' Our model isdescribed by a tight-binding Hamiltonian (with the pos-sibility of introducing disorder in the on-site energies)and uses a recursive Green's-function technique to calcu-late the resistance. ' As a simple illustration of theinfiuence of a junction on the transmission coefficients,

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we consider a three-terminal "T"junction without dis-order. The inset in Fig. 2(b) shows the fraction ofcurrent transmitted down the side probe and straightthrough when a single mode is injected down one branchof the main channel. The ground state propagates pref-erentially in the forward direction while the higher-lyingmodes have a large probability to turn the corner(though they do not necessarily remain in the samemode). Qualitatively, those modes turn the corner whosewave vectors closely match the wave vector of a mode inthe side probe.

In Fig. 2(b) we show AR as the current path bendsthrough a junction at a remote point (solid line), at onevoltage terminal (dotted line), and at each of the voltageterminals (dashed line). The structure is a main channelwith three cross strips without disorder. AR~ is averagedover energies with the same transverse channel number.AR generally decreases as the number of channels in-creases. The magnitudes correspond well with the exper-imental results given in Table I even though our simplemodel includes no material parameters! Theoretically,AR~ is expected to decay exponentially with scale L, asthe distance between the voltage terminals and the junc-tion where the current path bends increases since elasticimpurity scattering scrambles the relative mode popula-tions. Our estimate of L, = 5 gm = hp/e2zn; -z (forNo. 1) is obtained from the mobility which predominant-ly measures large-angle backscattering. The much morerapid decay observed experimentally (L, = 500 nm)might arise from small-angle forward scattering in theGaAs-A16aAs wires. ' '

Finally, the irrelevance of the impurity configurationto our results in the highest mobility devices is demon-strated in the inset of Fig. 2(a). The symmetric magne-toresistance changes little when an equivalent, butopposite-handed, bend is substituted in the current path

even though the bend is within L~ of the voltage termi-nals. Both the symmetric and asymmetric componentsof R3 ~o4s and R3s4$ are highly correlated [e.g. , 95%,80%%uo, respectively for the example shown in Fig. 2(a)].

R. Landauer, in Localization, Interaction, and TransportPhenomena, edited by G. Bergm ann and Y. Bruynseraede(Springer-Verlag, Heidelberg, 1985), pp. 38-50.

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4W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D.Jackel, D. M. Tennant, and A. D. Stone, Phys. Rev. Lett. 58,2347 (1987).

5P. M. Mankiewich, R. E. Behringer, R. E. Howard, A. M.Chang, T. Y. Chang, B. Chelluri, J. E. Cunningham, andG. Timp, J. Vac. Sci. Technol. B 6, 131 (1988).

J, E. Cunningham, W. T. Tsang, G. Timp, E. F. Schubert,A. M. Chang, and K. Owusu-Sekyere, Phys. Rev. B 37, 4317(1988).

7G. Timp, A. M. Chang, J. E. Cunningham, P. Mankiewich,R. Behringer, T. Y. Chang, and R. E. Howard, Phys. Rev.Lett. 58, 2814 (1987).

G. Timp, A. M. Chang, P. deVegvar, R. E. Howard, R. E.Behringer, J. E. Cunningham, and P. M. Mankiewich, Surf.Sci. 196, 68 (1988).

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).

'OM. Buttiker and Y. Imry, J. Phys. C 18, L467 (1985).'lM. Ma and P. A. Lee, Phys. Rev. B 35, 1448 (1987).12M. Biittiker, Phys. Rev. Lett. 57, 1761 (1986).' H. U. Baranger, A. D. Stone, and D. P. DiVincenzo, Phys.

Rev. B 37, 6521 (1988).'4S. DasSarma and F. Stern, Phys. Rev. B 32, 8442 (1985).'5F. F. Fang, T. P. Smith, and S. L. Wright, Surf. Sci. 196,

1988 (1988).

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Propagation around a Bend in a Multichannel Electron Waveguide