PHYSICAL REVIEW B VOLUME 51, NUMBER 4 15 JANUARY 1995-II

Noninteger conductance steps in a gapped double electron waveguide

Guangzhao Xu and Lin JiangDepartment of Physics, Fudan University, Shanghai 200433, People's Republic of China

Ping JiangChina Center ofAdvanced Science and Technology (8'orld Laboratory), P 0 .Bo.x 8730, Bejiing 100080, People's Republic of China

and Department ofPhysics, Fudan University, Shanghai 200433, People's Republic of China

Dong Lu and Xide XieDepartment ofPhysics, Fudan University, Shanghai 200433, People's Republic of China

(Received 8 June 1994)

A gapped double electron waveguide that contains a gap in the hard middle wall between two

waveguides is proposed, and transport properties of the gap, which is also an electron waveguide, are an-

alyzed. Noninteger conductance steps are found. Similar to what happens in a quantum point contact,conductance steps are found to arise whenever an additional propagation mode in the gap is excited asits width is increased; however, the contribution of each mode to the conductance is found to be less

than 2e /h due to the current branching at the gap. It has also been found that when the width of the

input waveguide is modulated, conductance for the current leaked out of the gap shows strong oscilla-tions lined up with the 2e /h steps of the input waveguide conductance, which can also be characterized

by current spectroscopy of the waveguide as del Alamo and Eugster suggested for a leaky electronwaveguide.

I. INTRODUCTION

In recent years, extensive attention has been paid tothe exploration of the wave nature of electrons in nano-structures. Demonstration of electron-wave guiding inhigh-mobility semiconductor heterostructures has notonly uncovered another regime of electron transport insolids, ' but will perhaps lead to enhanced functional elec-tron devices. In past years, electron waveguides withvarious configurations and structures have been studiedboth theoretically and experimentally. " Perhaps themost dramatic proof of electron-wave guiding is the ex-perimental observation of quantized 2e /h conductancesteps in a quantum point contact, which have beenviewed as short electron waveguides. ' ' The quantiza-tion of conductance is believed to be the result of thequantized transverse modes imposed by the confinementgeometry. ' The steps are found to correspond to theopening of propagation modes in the waveguide or, inanother sense, the crossing of subbands over the Fermilevel. Theoretical calculations have shown that in a per-fect electron waveguide, the conductance of a singlepropagation mode is 2e /h. ' Very recently, investiga-tions of the ballistic transport through a one-dimensional(1D) quantum system connected with a 3D electron reser-voir showed that the height of the conductance stepcould be (2e /h)n, ' where n is an integer other than uni-ty.

Recently Eugster and del Alamo implemented a leakyelectron waveguide, ' i.e., a waveguide with a thin sidewall. While the electron waves propagate along thewaveguide, the electrons could partially leak out of it bytunneling through the thin wall. They argued that the

A

II

C A'

B B'

FIG. 1. Illustration of the theoretical model adopted to simu-late the gapped electron waveguide.

leaked current could be characterized as a current spec-troscopy to study the 1D density of states of thewaveguide. Their suggestion has been confirmed by theauthors of the present paper through a detailed theoreti-cal investigation. ' Based on these experimental andtheoretical studies, it is naturally logical to ask what will

happen if electrons do not leak out of the waveguide bytunneling through a thin side wall but through a window,or a gap, on a hard wall. This is just the motivation forthe present paper.

Our model structure is shown in Fig. 1. Two electronwaveguides A-A' and B-B' are separated by a hard wallof thickness t between them, and a window or a gap ofwidth d is assumed to be present on the wall. The twowaveguides could not couple to each other except

0163-1829/95/51(4)/2287(4)/$06. 00 2287 1995 The American Physical Society

2288 XU, JIANG, JIANG, LU, AND XIE

through the gap. It is assumed that electrons arelaunched into waveguide A, which is connected at oneend (say, the left end) to the input electron reservoir,which is not plotted in the figure. While the inputtedelectrons travel down waveguide A-A', part of them willleak out through the gap to waveguide B-B'. Obviously,the gap can also be regarded as an electron waveguidealong the y direction, and is called a gap waveguide here-after. Calculations show that in this structure the contri-bution to the conductance of each propagation mode inthe gap waveguide is less than 2e /h, due to the inputcurrent branching at the gap. Besides, when the width ofwaveguide A-A ' is swept (without modulating the widthof the gap waveguide), a conductance oscillation of thegap waveguide appears in line with the conductance stepsof waveguide A-A'. This oscillation can be regarded asthe same as in a leaky electron waveguide observed exper-imentally by Eugster and del Alamo' and analyzedtheoretically by the present authors, ' and can also beused as a current spectroscopy of waveguide A-A '.

In principle, summations in Eqs. (1)—(3) should runover all the modes in corresponding waveguides, includ-ing both propagation modes whose transverse energiesare lower than Ez and evanescent modes whose trans-verse energies are higher than Ez. In the real calcula-tion, however, taking all the propagation modes and alimited number of lowest evanescent modes into accountwould meet the desired precision.

Boundary matching at the interfaces of regions I andII (x =0), and regions II and III (x =d) gives all thecoefficients in Eqs. (1)—(3) and thus determines the wavefunctions. Conductance for the current Aowing downwaveguide A ' corresponding to the qth input mode canbe written as

n A

and the conductance for the current leaked out throughthe gap is

II. THEORETICAL MODEL (6)

The model structure shown in Fig. 1 is divided intothree regions, i.e., region I for x &0, region II for0&x &d, and region III for x)d. Electron wave func-tions in the three regions (denoted as f„f», and g», ) canbe expressed as

—ik xQ(x,y)=P~ (y)e " +pa P" (y)e

+g b„g~(y)e

Here we summed up the currents Aowing forward inwaveguide B' and backward in waveguide B to obtain theleaked current. n in Eqs. (5) and (6) runs over all thepropagation modes in waveguides A-A' and B-B', re-spectively. Considering the multimode input, the totalconduciances are

GD=X GS

ik'x —ik'xg»(x, y) =g (c&+e +c& e )Pc(y),

1

Att(xy)=X~.'4~(y)e " +X~.'PWy)e '(2)

(3)

Gl =g Gg,

where summations run over all the propagation modes inwaveguide A.

III. RESULTS AND DISCUSSIONS

/2k n2

+E;n =EI;,2m

(4)

where E;" is the transverse eigenenergy of the nth modeand EF is the Fermi energy of the electron reservoir.Equations (1)—(4) are for a single input mode. To obtainthe conductance of the device, contributions from all theinput modes below the Fermi level should be summed up.

respectively. In the above expressions, e' represents thewaves propagating forward along the x direction and

ikgq xe ' the waves propagating backward. P~z (y)e inEq. (1) is the electron wave injected in waveguide A.P,"(y) is the wave function of the nth transverse mode ofwaveguide i, where i can be A, B, C, A', and B'. In Fig.1, region I consists of waveguides A and B, region IIIconsists of waveguides A' or B', region II is waveguideC. Obviously P z =P„., Pz =Pz . Assuming theconfinement of all waveguides is of the hard wall type,P,". (y) can simply be written as sinusoidal functions, k;" isthe wave vector of the nth mode in waveguide i, whichshould satisfy energy conservation:

In the present paper we have used normalized units.The length is in units of 8' an arbitrarily chosen length.Correspondingly, wave vector k is in units of n/W andenergy of A' n. /2m *W .

Figure 2 shows the decreasing steps of the waveguideconductance GL, and increasing steps of leaked conduc-tance GL versus d, the width of the gap waveguide. Ac-tually the gap waveguide is a part of waveguide C, butviewed in a diFerent direction. Waveguide C is along thex direction, while the gap waveguide is along the y direc-tion; therefore the length of waveguide C (d) is the widthof the gap waveguide. The two series of steps appear at dvery close to 0.20, 0.40, 0.60, and 0.80. We have assumed8'~ = 8' therefore the threshold energy of the nth modein waveguide A is n in normalized units. GivenEF =26.0, five input modes are excited in the inputwaveguide A. It is easy to see that in the normalizedunits d=s/QE~ is the necessary width to open the sthmode in the gap waveguide, and the above values of djust correspond to s = 1, 2, 3, and 4, i.e., to the opening ofthe first, second, third, and fourth modes of the gap

GAPPED DOUBLETANCE STEPS INNONINTEGER CONDUCT 2289

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0.30

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0.85

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0.0

d=0.50

0.4

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diFerent values of gapFICx. 4. GL vs the gapa length t for 1 erwidth (d).

2290 XU, JIANG, JIANG, LU, AND XIE 51

satisfy 2kL ht =2~. Energy conservation gives

kL =vr+Ez —1/d . Thus we have b, t= 1/&10=0.31,which agrees quite well with the result shown in Fig. 4(a).For d=0. 50, two modes are excited (in this case, theopening energy threshold for mode 1 and mode 2 is 4.0and 16.0, respectively) and the oscillation turns out to bemore complicated as shown in Fig. 4(b).

Moreover, sweeping the width of waveguide A ( W~ )

also reveals very interesting results. As shown in Fig. 5,leaked conductance shows strong oscillations in line withthe 2e /h steps of the waveguide A-A' conductance(GD ). Similar features have been experimentally ob-served by Eugster and del Alamo' and theoreticallydemonstrated by the present authors. ' However, in pre-vious structures, i.e., in a leaky electron waveguide,current leaks out of the waveguide by tunneling througha controlled potential barrier, instead of a gap on thehard side wall of the present paper. With an approximatetheory [Eq. (6) or Ref. 18], the oscillations are explainedas a result of the 1D density of states sweeping throughthe Fermi level. To explain the present results, Eq. (6) ofRef. 18 can still apply except that T is no longer a con-stant while modulating Wz, since the scattering near thegap is more complicated than simply tunneling through athin wall. Therefore, detailed features of the oscillationswould be different from those observed by Eugster anddel Alamo, as one could see that the oscillation ford=0. 5 in Fig. 5 looks different. However, the effect ofthe addition of one propagation mode that gives rise to adifferent oscillation of the leaked conductance as well as adifferent 2e /h step of waveguide 3-2 ' conductance stillremains unchanged. Therefore, the proposed structurecould also be designed to study the current spectroscopy

6.0

5.0

4.0

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1.0

EF=10.0 t=0.5 WB=4.0

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0.6 p0.4

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0.02.0

of a waveguide.In summary, a gapped double electron waveguide is

proposed and its transport properties are studied from atheoretical approach. Noninteger steps of the conduc-tance for the current leaking out through the gap arefound. Oscillations of the gap-waveguide conductancewhile sweeping the width of the input waveguide are alsodemonstrated, which are explained by comparing withthe tunneling spectroscopy of a leaky electron waveguide.

FIG. 5. Waveguide conductance (GD) and leaked conduc-tance (GI ) vs the width of waveguide A (8'&) for differentvalues of the gap width (d}.

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