Extended Abstracts of the 1992 International Conference on Solid State Devices and Materials, Tsukuba, 1992, pp. 753-755

Transport Properties of Ballistic Quantum Wire

Toshihiro Itoh, Nobuyuki sano, and Akira yoshii

NTT LSI Laboratories3 -I Morinosato Wakamiya, Atsugi- shi,

Karwgawa -ken 243 -0 I, JAp AN

s-ill-15

small enough has an effect similar to intrroducing attactiveimpurities. Recently, the authors investigated a structurcsimilar to that of Kasai et al. and clarified that thequasibound states arise as a result of mode couplingsand that these states are localized around the region ofthe width increase.t) Ail these pictures are restriited tothe case when the propagating modes can be definedapproximately throughout the wire.

In the present study, we investigate the trans.portcharacteristics of ballistic quantum wires with a singlestub where the above pictures no longer hold true. Inthis case, the modes mix strongly and they cannG bedefined throughout the wire. It is shown that thequasibound states also exist in these wires and thesestates are closely related to the transport properties ofthe wires in the structure with rounded corners, as wellas in the strucnrre with sharp corners.

2. Method

We employed the tight-binding Green functionmethod, which is a slight modification of the recursiveGreen function method.e) Conductance and local densityof states per unit length were calculated and compared.This approach has been recently proposed by the authorsfor the analyses of ballistic quantum wires. Thetransmission matrix can be determined by the Greenfunction and the Landauer formula is used to evaluatethe conductance. The local density of states per unitlength of the wire at rx, D* (r*, E ) , and the localdensity of states per irnit lenlth for modes at !,P^(r*,m,E ) , are defined as

We investigqte electronic properties of balli*ic quantum wire with a stub by the tight-binding Green function methd. Ii is ihown that well-defined quasi-bound rtato are formed inthis structure and that the dip structure in the conductance reiults from these rtut"r. A morerealistic structure with round corners is also analyzed and there is no qualitativ" diff"r"ncebetween the results for the two structures, especialty in the single mode regi-".

L. fntroduction

The recent progress of microfabrication techniqueshas made it possible to crcate a nanometer-scale strucnuein semiconductors. The characteristics of this strucmrereflect the wave nature of electrons. Among thesecharacteristics, the conductance of the qoantuit pointcontact in the modulation-doped heterostructure has-beenfoundp show step stucture as a result of the quantizationof the transverse momentum.l)

Using this quantum wire structure, many kindsof devices that utilize the interference effects have beenproposed. sols et al.2)'3) proposed a quantum modulatedfransistor which is a quantum wire with a single stub.This device features a singly connected quantim wireand the cunenr is modulit6a uy a remore gate. Thiswire has a dip structure in the conductanci and thecurrent is modulated by controlling the position of thedips.However, as the physical picture of this dip structureremains unclear, T4ysis of experimental daia relatingto this effect is difficult.a) Moreover, it is not cleaFwhether this effect remains when the abrupt corners arerounded, as,the abrupt corners do not e*ist in the realnanometer-scale structures.

Similar dip structure has been reported byBagwells) and Telqnan et al.6), both of whom analyzeia system in which atractive impurities were inroducedi"^F: tr, 3"y suggested that the bound srares splittingoff from higher-lying confinement subband aito tttepropagating states in the lower mode couple to becomethe quasibound states and totally reflecting states, andthat these states are the source or *re dip itructures inthe conductance. Kasai et al.?) investigated a similarstructure with a slightly increased width, which when T

fr,Itv

753

D*( r*E ) - p(7, E) = P ^( rx, m ,, E), (1)

50nm

Fig. 1. Abrupt T-stub sffucture.

P^(r*m,,E )= ?

A( t - n).gn( r,,miA\(r,,n y) , e)

en(r,, ^ r) = lf ^,( rr)Q o(r,,r y) , (3)

where Q^( t*) and { are the complete set oforthonormal eigenfunctions and eigenvalues of theHamiltonian H, respectiv ely . f

^ "(r r) is the eigenfunction

with respect to the transverse mode mn and p ( t=, E ) isthe local density of states per unit area. In the tight-bindingmodel,

f^,(rr)= J;=sin ( #), (4)

where N, is the number of sites in the transverse

direction. p6 (rr:, mn, E ) can be directly determinedfrom the imaginary pan of the diagonal element of the

Green function. p* (r* mn,E ) can be considered to be

the decomposition of the lbcal density of states D, (r*,E ) for transverse momentum. The effective mass ofGaAs (0.067m0, where mois the electron rest mass) is

used and the lattice constant of the tight-binding latticeis taken to be 2.5nm throughout this study.

3. Results

First, the conductance I of abrupt sffucture as

shown in Fig. 1 is plotted in Fig. 2 (a). The profilewidely deviates from the conductance quantizationprofile, which implies that the modes mix strongly atthe junction. The dip structure is seen in the profile.The local density of states D* (r* E ) at C in Fig. 1 isplotted for (Fermi) energy in Fig. 2 (b).It is clearlyseen that for each dip in the conductance profile, thelocal density of states has a clear sharp peak. Thecorrespondence is especially clear for the dips in thesingle mode regime. As the peaks have small widths,they can be regarded as quasibound states. To clarifythe origin of these peaks, Dn (r* E ) was decomposed

to pm (rr, mn, E ), which is plotted in Fig. 3 (a). It canbe seen that inost of the peaks have well-defined modeindices corresponding to tlre discrete levels of the quantumbox in the stub region. The box is indicated by thedot-dashed lines in Fig. 1. The second, third, and fourthpeak in Fig. 3 (a) correspond to the states which havethe mode indices (rflr,m) = (1,3), (1,4), (1,5),respectively. The first peak cdrresponds to the state whichis the combination of (1,1) and (1,2). The fifth peakcorresponds to the state which is the combination of(3,1) and (3,2). The peak for m, = 2 cannot be found

in this profile, because p^ (r* mn, E ) is plotted at C

0 5 l0 15 20 25 30Energy (meV)

Fig. 2. (a) Conductance g, O) local densiry of states perunit length Dn (rr, E) at C of the abrupt structurc.

0510 15202s30Energy (nteV)

Fig. 3 Local density of states per unit length for

Tod"r p^(rr,^u,E) at (a) C, (b) Q, and (c) L of theabrupt structure.

d

3

o-

754

50nrn<-

1L

clcloolFig. 4. T-stub structure with rounded corners.

where there is a node. This is confirmed by Fig. 3 (b),in which p*(rr,m,E ) at Q in Fig. I is plotted. A newpeak appears at a6out E = 9 meV and this correspondsto the third dip in Fig. 2 (a).

As a result of attaching the leads, the discretelevels in the quantum box have finite probabilities ofdecaying outside the stub and they become the quasiboundstates. The positions of the peaks are slightly shiftedfrom the discrete levels as a result of the couplings. Thewidths of these peaks can be related ro the lifetime ofthese quasibound states by the uncertainty principle.

for comparison, pa(r*,mn,E ) at L in Fig. 1 isplotted in Fig. 3 (c). Here-a {uasi-one dimensionalsubband spectrum can be seen and it is totally differentfrom the spectrum in Fig. 3 (a) or (b). The oscillatorystructure is the result of the interference effect.

Real systems fabricated by etching or the lift-offprocess never have the sharp corners as shown in Fig.1. Accordingly, the four corners were rounded as shownin Fig. 4 to investigate the robustness of the resultsobtained above. The conductance g and the local densityof states for modes p^(r*,mn,E ) are plotted in Figs. 5(a) and (b). The rounding cduses no qualitative changein the result in the low energy region. Peaks in the localdensity of states are enhanced or reduced and slightlyshifted. The peak at around E = 22meV disappears. Itmeans that this rounding destructs only the quasi-boundstate which has mode index ffi- = 3. This is reasonablebecause the states which havetmal mode indices havelarge wavelengths and are not severely affected by therounding of the corners. It is thus expected that this dipstructure is not affected qualitatively as far as the well-defined quasibound states are formed in the stub region.

4. Conclusion

The electronic properties of a ballistic quantumwire with a single stub has been investigated by thetight-binding Green function method. The physical originof the dip structure in the conductance profile wasclarified; there are well-defined quasibound states andthey correspond to the discrete levels of the quantumbox. The conductance shows a dip structure at the energycorresponding to these quasi-bound states. A morerealistic structure with rounded corners was alsoanalyzed. No qualitative change in the results for theabrupt structure was found even after the corners wererounded, especially in the single mode regime. Thecharacteristics of quantum interference device are closelyconnected with their structure, so our investigation givesan important insight into the designing of these devices.

0 5 l0 15 20 25 30Energy (meV)

Fig. 5. (a) Conductance-g of rounded (solid line) andabrupt srructure (dotted line). (b) local brnrity or'stut sper unit length p*(r,,m, E ) of rounded sffucture.

Acknowledgments

The authors would like to thank M yamamotoand K. Aihara for helpful discussion and K. Hirata forhis encouragement.

References

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4)-Recently, the interference effect in this structure (with alonger stub) was. reported by K. Aihara, M. yamamoto,and T. Mizutani,.Jpn. J. Appl. phys. (ro be published).They used one-dimensionai-picture for atraiyring tneexperimental data.

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published)

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