evanescent modes and scattering in quasi-one-dimensional wires

PHYSICAL REVIEW B VOLUME 41, NUMBER 15 15 MAY 1990-II

Evanescent modes and scattering in quasi-one-dimensional wires

Philip F. BagwellDepartment ofElectrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 12 January 1990)

We calculate the current transmission amplitudes and electrical conductance as a function of Fer-mi energy for electrons scattering from a single defect in a quasi-one-dimensional wire. In aconfined geometry the scattering boundary conditions couple propagating modes in the wire tononpropagating or evanescent modes. Therefore, the applied steady current causes localized orevanescent modes to build up around any defects in the wire. These extra stored electrons stronglyaffect the scattering boundary conditions for the propagating modes whenever the Fermi energy ap-proaches either a new quasi-one-dimensional subband or a quasi-bound-state splitting off of thehigher confinement subbands. We show that the presence of evanescent modes can lead to eitherperfect transparency or perfect opaqueness for the scattering modes, even in the presence of scatter-ing defects. For the special case of a 5-function scatterer in the wire we analytically obtain thescattering amplitudes. We also numerically examine a finite-range scatterer.

I. INTRODUCTiON

When an electron scatters elastically from an imperfec-tion in an open geometry, such as the scattering from apotential-energy barrier or well in an infinite three-dimensional space, it scatters into a traveling wave whichpropagates away from the defect. In contrast, if the elec-tron is restricted to a wire such that confinement sub-bands are formed, the incident electron can elasticallyscatter into evanescent modes available in the wire. Thusfor a steady current fiow incident on a defect in the wire,a localized mode will build up around the defect even ifthe scatterer is repulsive. Unlike the case in electron lo-calization theory where the "localization" is either aconsequence of the coherent scattering of electrons frommultiple defects or of electrons trapped in potential-energy wells, the localized electron trapped around a sin-

gle scattering defect in a narrow wire is maintained thereby the applied incident current, regardless of the sign ofthe scattering potential.

For the special case of an attractive scatterer in a nar-row wire, it is possible to have quasi-bound-states split-ting off from one 'of the higher-lying confinement sub-bands. The bound state associated with the lowest sub-band is analogous to a donor level below the conduction-band minima of a semiconductor. An electron trapped inthe lowest subband's "donor level" at zero temperaturecannot escape the region of space near the donor; it istruly bound. But since the confinement potential of thewire gives rise to multiple subbands, there will be a new"quasi-donor-level" associated with each subband. Theseadditional "bound-state remnants'* are also spatially lo-calized near the defect and composed of evanescentwaves, but are not true bound states. Because they aredegenerate in energy with propagating modes in the wireand are coupled to the propagating modes, they will de-

cay with time. But in contrast to the unbound evanescentwaves which can build up around either an attractive or arepulsive scattering defect, we believe the quasi-bound-states can persist for much longer times if the incidentcurrent is reduced to zero. Analyzing how these newquasi-bound-states will decay with time is an interestingproblem in its own right, but for the purposes of this pa-per it is sufficient to realize that these states exist and areimportant for electron transport in confined geometries.

We show in this paper that the presence of evanescentmodes strongly affects the scattering of propagatingmodes from a defect in a quasi-one-dimensional wire.For interacting electrons this is clearly true because anelectron trapped near a scattering defect will alter thescattering potential near the defect. But even for thenoninteracting electrons which we consider in this paper,the building up of evanescent modes near the scatteringdefect alters the boundary conditions for the scatteringevent. This leads to unusual scattering properties whenthe Fermi energy approaches either a subband minima ora quasi-bound-state splitting off of a higher-lyingconfinement subband, so that evanescent modes cannotbe neglected when analyzing scattering in a confinedgeometry. We are careful in our analysis to consider theeffects of both propagating gnd evanescent modes duringthe scattering event. It is well known from the analogouscase of electromagnetic wave scattering in microwavewaveguides that one cannot neglect the electromagneticenergy stored near any defects or sudden spatial varia-tions in the guide when calculating its scattering proper-ties. For optical waveguides one must also consider elec-tromagnetic radiation generated at defects, an effectwhose analog is absent in our problem.

Qur main goal is to understand the simplest possiblescattering problem in a quasi-one-dimensional wire:scattering from a single 5-function defect in an infinitely

41 10 354 1990 The American Physical Society

41 EVANESCENT MODES AND SCA i iERING IN QUASI-ONE-. . . 10 355

long wire. We analyze the problem following the modematching method outlined by Datta et al. ' and Cahayet al. , and which has been popular for calculating thescattering properties of microwave waveguides. For thesimple case of a 5-function scatterer, the varioustransmission and reflection coefficients can be obtainedvery straightforwardly by hand for any number of propa-gating or evanescent modes. We also present numericalresults for the transmission coefficients through a finitesize rectangular obstacle in a quasi-one-dimensional(Q1D) wire. For the case of a rectangular obstacle, thereare two sets of subbands which are important in thescattering process and the effects of both can be seen inthe transmission coefficients and the electrical conduc-tance. For the finite size barrier, we obtain the varioustransmission and reflection coefficients by cascadingscatter matrices as outlined in Refs. 1 and 2. We showthat the transmission as a function of incident electronenergy is in general enhanced at each subband minimumand suppressed at a quasi-bound-state.

Most theoretical calculations of the conductance ofQ1D wires model the impurity potential using an Ander-son tight-binding Hamiltonian, so that the problem is putonto a lattice, the required Green functions are numeri-cally obtained using a recursion method, and from themthe conductance of the wire is calculated. This recur-sion method has been used to discuss the Hall effect innarrow wires. A more conceptually straightforwardmethod of simply matching wave functions at the bound-ary of a Hall probe and using a computer to solve the setof simultaneous equations has also been applied tothis problem. The recursion method " and wave-function matching method' have been used to study con-stricted geometries with and without disorder. If theconstriction forms a "saddle" potential, one can also ob-tain the transmission coefficients analytically. ' Insteadof these more complicated geometries, we choose to focuson the simpler case of a single obstacle in a wire of uni-form width, so that we may either calculate the scatteringamplitudes analytically or by numerically cascading asmall number of scatter matrices. After calculating thescattering amplitudes, we obtain the conductance fromLandauer's formula, ' using both the two-probe andfour-probe versions of Landauer's formula. ' The de-velopment of various Landauer conductance formulas infact motivated all of the work in Refs. 1—13. For two re-cent reviews discussing this progress see Refs. 21 and 22.

Our final results resemble those of Chu and Sorbello,who used a completely different method, based on ananalogy with the method of images in electrostatics ap-plied to the propagating modes, to solve this problem.They considered an s-wave scatterer of various strengthsand phase shifts, while we consider a definite scatteringpotential. Reference 23 attributes the behavior of theconductance versus Fermi energy to an interaction be-tween the scatterer and the waveguide walls, while in ourformalism it is a consequence of evanescent modes creat-ed in the wire due to the applied incident current. Ourresults for an attractive scatterer also resemble some ofthose obtained by Masek et al. , so that we choose to at-tribute their results to a quasi-bound-state appearing in

their calculation. Similarly, we attribute the behavior ofthe conductance for the attractive scatterer studied inRef. 23 to the appearance of a quasi-bound-state.

II. SCATTERING IN QID WIRES

d, + V, (y} Z. (y»}=E.X.(y}

2nl dy(2)

where n is the subband index and E„arethe subband en-

ergies. We will expand the solutions to the full scatteringproblem of Eq. (1) in terms of the basis set y„(y)whichare solutions of Eq. (2}.

For each position x along the wire, we can write thedependence of the wave function along y as a sum over aset of Fourier coefficients c„(x)as'

ttr(x, y) = g c„(x)y„(y). (3)

Equation (3) must hold simply from the competenessproperty of the y„(y}'s.Substituting the wave functionP(x,y) from Eq. (3) into the Schrodinger Eq. (1), we ob-

Incident Xr(y) Transmitted Xr(y)

sr = )W--

x=0I

x=, LI

I

I

I

~l ~ i'--f/~ gL ~- &~—-~

0 =x

EN(k)

Mode Xe(y)

-EF=k

FIG. 1. A single scattering defect in a quasi-one-dimensionalwire. The wire is assumed to be infinitely long on either side ofthe defect, and to have adiabatic connections to infinitely wide

regions (reservoirs not shown) at x =+00 as in Ref. 19. For car-riers incident only in the lowest subband as shown, evanescentwaves build up on either side of the defect in the second andhigher normal modes.

Consider a quasi-one-dimensional wire having elec-trons confined along the y direction but free to movealong the x direction as shown in Fig. 1. The fullSchrodinger equation of motion is

g2 d2 d22

+2

+ V, (y)+ V&(x,y)dx dy

X f(x,y) =Ep(x,y) (1)

where the confinement potential V, (y) depends only onthe transverse direction y and Vz(x,y) is the potential ofany defects or impurities in the wire. If we had con-sidered a one-dimensional problem along y in the regionswhere there are no defects, the confinement potentialwould give rise to a set of normal modes y„(y)satisfyinga one-dimensional Schrodinger equation such that

10 356 PHILIP F. BAGWELL

d c„(x)+k„'c„(x)= y r„.(x)c.(x)dx

(4)

where the wave vector k„is

k„= (E E„—)

and the I „'sgiven by

tain an equation of motion for the Fourier coefficients

c„(x)as'

(k )'~' (k )'" CJ t J

(k )1n ' J (I )&n(12)

where we have defined t; =C /A; as the ratio of thewave-function amplitudes. Similarly, the reflectioncoefficients R„are

when all other incoming currents are zero except those inchannel i. Here t, is the current transmission amplitudeas defined in the scattering matrix. In terms of thecoefFicients A„,8„,C„andD„these are

I'„(x)= dy X„*(y)Vd(x, y)X (y)2777

$2(6) J kj left j

i, left i(13)

are the mode coupling constants. Equation (4) allows anynormal mode to scatter elastically into any other normalmode through the defect potential Vd(x, y). One musttherefore solve a scattering problem with an infinite num-ber of coupled modes to obtain the exact answer for thescattering properties of any one normal mode. The sumon the right side of Eq. (4) includes both the term n =mand the coupling to any evanescent modes for which weset k„=ix„.

The boundary conditions for solving this scatteringproblem are analogous to the ones from textbookquantum-mechanical scattering problems in one dimen-sion. Continuity of the wave function 1(t(x,y) requiresthat each Fourier coefficient c„(x)be continuous. If thescattering potential Vd(x, y) is nonsingular, then it fol-lows by integrating Eq. (4} and using the continuity of thec„(x)'sthat the derivative dc„(x)/dx must be continuousfor each n. We discuss the boundary conditions forsingular potentials in the next section.

In the regions where the scattering potential is zero,namely, regions I and III as shown in Fig. 1, the solutionsto Eq. (4) are

ik„x —ik„xA e "+Be ", x&0

where r, =8 /A; is the ratio of the re6ected wave-function amplitude to the incident amplitude. The two-probe current at small voltages we obtain from one ofLandauer's formulas'

2 2 e 2

i,j l,J

(14)

where the sum in Eq. (14) runs only over the propagatingnormal modes of the wire. Equation (14) and its generali-zations to finite voltages are equivalent to standard tun-neling conductance formulas. The quantum contactresistance implied by Eq. (14), discussed in Refs. 17—22,has been seen in a striking and clear fashion.

We shall also examine the four-probe Landauer con-ductance. ' ' ' ' ' For the cases we consider in this pa-per, an electron incident on the scatterer from either endof the conductor will have the same transmission proba-bility. Or in the notation of Ref. 15, we will have

Tj Tj and R 'j R 'j in all our calculations. For thiscase the four-probe conductance can be written moresimply as

c„(x)= ik„(x—L) —ik„(x—L) I eGfour-probe Vi g X Tij QNi

I& J

for the propagating modes and, by setting k„=ix„,ob-tain

X gN; QR,"l J

(15)

A e "+B„e",x(0c„(x)=. —K (x —L) K~ (x —L)Ce " +D e", x)L, (10)

for the evanescent modes. Solving Eq. (4} with the ap-propriate boundary conditions will determine thecoefficients A„,8„,C„,and D„in each region. We de-scribe a method for doing this in Sec. III and V.

Suppose the coefficients A„,8„,C„,and D„arenowknown. Define the current transmission probabilityT j Tj from normal mode or "channel" i on the leftto channel j on the right following the diagram inButtiker et al. ' by

j,rightJLJ

Ji, left

Equation (15) is just the two-probe conductance from Eq.(14) multiplied by a voltage division factor. The currentsI in Eqs. (14) and (15) are the same but the voltages V andV' are measured differently. ' In Eq. (15) N, is the one-dimensional free-electron density of propagating statesfor the ith occupied subband. Also following Ref. 15, weassume that the weakly coupled voltage probes are farenough away from the scatterer so that evanescent modesexisting near the voltage probes can be neglected. Thisassumption is always violated near the subband minima.However, for our special case, the evanescent mode am-plitude is the same on either side of the scatterer so thefour-probe formula of Ref. 15 is likely valid for our prob-lem despite its neglect of charge accumulation near thevoltage probes in the evanescent modes.

If evanescent modes are important in any scattering

EVANESCENT MODES AND SCA I I'ERING IN QUASI-ONE-. . . 10 357

Eg

Density of PropacIating Statesso that we take I. =0 in Fig. 1. The weight y can be ei-ther positive or negative. Integrating Eq. (4) across the 5function gives

Ep=E

Ep

dc„(x)dx

dc„(x)dx x=0

(18)

EN(K) Density o& Evanesce&t Stateswhere

E (19)

E2 Eg

FIG. 2. Density of states in a quasi-one-dimensional wire. (a)shows the unusual dispersion relation and density of propaga-ting states while (b) emphasizes that evanescent states also existin the wire. Thus, in the presence of scattering defects, thestates available in (a) are not the only relevant ones to consider.The evanescent density of states depends on position as ex-plained in the text.

ik„(C„D„)i—k„—(A„—8„)=g I „(A+8 ) (20)

if n denotes a propagating mode and, setting k„=ix„,be-comes

are the mode coupling constants. The I „arepropor-tional to the strength of the impurity and the size of thewave function at the impurity, and are zero if the wavefunction has a node at the impurity position. Equation(18) reduces to

problem, as they must be for scattering in a confinedgeometry, then the position-dependent density of states

—v„(C„D„)+tc„—( A„—8„)= g I'„(A +8 ) (21)

N(E;x,y) = g 5(E E„„)Ig„k—(x,y)I'n, k

(16)

will no longer be just the usual result for clean wireshown in Fig. 2(a}. Instead, for positions near the defect,there will be an extra piece added to the position-dependent density of states which will resemble Fig. 2(b).Figure 2 again emphasizes that evanescent modes willhave an effect on the scattering properties of the wire wellbefore the subband minima is reached. If quasi-bound-states form in the wire for the case of an attractivescatter, then Fig. 2(b) will of course become more compli-cated.

The remaining sections will be concerned with solvingthe infinite set of coupled equations (4} for the wave func-tions c„(x),obtaining the various transmission andreAection coeScients, and studying the Landauer conduc-tance of the wire.

III. CASE OF A 5-FUNCTION SCATTERER

In this section we consider electron scattering from a5-function potential in a quasi-one-dimensional wire. Wewrite down the detailed analytic solutions to the infiniteset of coupled equations, Eq. (4), for the transmissioncoefficients T;. and reflection coefficients R; in AppendixB. These solutions are valid for 5-function potentials ofarbitrary strength and for any arbitrary number of propa-gating and evanescent modes. In this section we formu-late the problem and discuss some of its most interestingfeatures.

Let the 5-function scattering potential be

Vd(x, y) =y5(x)5(y —y; )

if n denotes an evanescent mode. The sum over m againincludes the term where m =n as well as all the evanes-cent modes. In addition, the boundary condition that thewave function itself must be continuous, c„(0)=c„(0+),gives

A„+B„=C„+D„ (22)

for all normal mode indices n.Imposing the remaining boundary conditions from our

scattering problem, proceeding as in the normal textbookquantum-mechanical scattering problems in one dimen-sion, allows further simplification of Eqs. (20), (21), and(22}. We consider particles incident only from the left.Then D„=Ofor all propagating n. Because we take thewire to be infinitely long, we can eliminate all growingevanescent waves and set D„=O and A„=O for allevanescent n. This leaves the condition B„=C„for allevanescent n and A„+B„=C„for all propagating n.Furthermore, for all the propagating A„'s,we allow onlyone normal mode incident at a time so that we have onemore unknown than equations (as is usual in scatteringproblems). We solve for all the transmitted and reflectedwave amplitudes with respect to the single incident A„.

We now find a procedure for solving the infinite set ofcoupled equations (20), (21), and (22). Truncating theequations to a finite size will allow us to find the rightprocedure. For definiteness, let us write down the matrixequation we must solve if we consider two propagatingmodes and two evanescent modes. Let mode two be in-cident on the scatterer so that A, =0 and B,=C, . Wewrite down the set of four coupled equations va1id whenE, &E (E3 as

10 3S8 PHILIP F. BAGWELL 41

—2ikz Izi I zz—2ikz I 23

I &4

I 24 t22

I4&

I 32

I 42

I 33+2K3

I 43

tZ3

I 44+ 2K4 tz4

(23)

Now let us eliminate the highest evanescent mode, in this case mode four, from the 4X4 matrix Eq. (23). We obtainthe 3 X 3 matrix equation

0 I )) 4—2ik, I &2,4 I i3,4 tz&

—2ikz I 22 4—2ikz (24)t22

I 3&,4

I 23, 4

0 I 32 4 I 33 4+ 2 t23

which has the same form as the matrix equation we would have started with if we had originally decided to allow twopropagating modes and only one evanescent mode. Equation (23) looks formally like Eq. (24} if we simply truncate thehighest evanescent mode in Eq. (23). The only difference between Eq. (24) and truncating Eq. (23} is that all the modecoupling constants in Eq. (24) have been rescaled by the evanescent mode four where

2K4I; 4 I; l,J =1,2, 344 K4

Now eliminate the next highest evanescent mode, mode three, from Eq. (24). We obtain

(25)

0—2ikz

I„—2ik, .

I zi, 3 —4

I &2, 3 —4

I 223 —4

tz,(26)

Again, this is the same matrix equation we would haveoriginally written down if we had allowed only the twopropagating modes and completely neglected the evanes-cent modes in Eq. (23}, except that the mode couplingconstants [which were initially only rescaled by modefour in Eq. (24)] are now rescaled by both the evanescentmode three and mode four as

2K3I =I"tj3 —4 ig4 I +2»J

4 K3(27)

The total effect of the evanescent modes is to rescalethe mode coupling constants of the propagating modes.The rescaling procedure makes the mode coupling con-stants energy dependent. And although we have writtendown this procedure only for two propagating modes andtwo evanescent modes, by induction it clearly holds forany number of propagating and evanescent modes. Ifthere are p propagating modes and q evanescent modes,we can eliminate the q evanescent modes by the pro-cedure outlined above and solve a p Xp matrix equationto obtain the transmission and reAection amplitudes forthe propagating modes, and hence determine the conduc-tance. We solve this matrix equation explicitly for thelowest three subbands in Appendix B and assert that, be-cause of the symmetry of the matrix equation analogousto Eq. (23), we can solve the p Xp matrix equation for anyinteger p. Hence we find a complete solution to ourscattering problem. We emphasize at this point that ourscheme for including the total effect of the evanescentmodes as a simple rescaling of the mode coupling con-stants for the propagating modes is only valid for the spe-cial case of a single 5-function scatterer in a narrow wire.

Clearly we must truncate this infinite set of coupled

I

equations in order to solve them, but how many equa-tions must we include in order to obtain a physicallycorrect answer? The above rescaling procedure tells usthat it depends on the strength of the scatterer. FromEq. (19) all the mode coupling constants are of orderI'„-2my/Wfi so that for strong scatterers the largerwill be I „„andhence it will be necessary to include moreevanescent modes in order to obtain the correct transrnis-sion probabilities. For evanescent mode n, the rescalingof the mode couplings constants will have negligibleeffect when the evanescent density of states times thestrength of the scatterer is small, (m/R ~„)(y/W)&(1,and so it is necessary to include enough evanescent modesin the calculations such that this inequality is satisfied.In real wires, which have a potential well of finite depth,evanescent states from the continuum may play some rolein determining their scattering properties if the scatteringpotential is strong enough.

Our mode rescaling procedure tells us another impor-tant fact: The modes completely "decouple" when theparticle's energy aligns with the bottom of a subband,and perfect transmission results for each of the separatemodes. Consider the case of two propagating modes de-scribed by Eq. (26}. From Eq. (27), as we approach thebottom of subband 3, we must rescale all the mode cou-pling constants by the factor 2~3/(I 33+2a3). This factorapproaches zero since K3~0 at the third subband mini-ma. Hence all the mode coupling constants become zero,making it appear as though the scatterer were absentfrom the wire as we approach each new quasi-one-dimensional subband. This "perfect transparency" effectcan be seen here for Tzz=l and Tz, =0 by substitutingthe mode coupling constants near the third subbandminima into Eq. (26). A similar equation written down in

41 EVANESCENT MODES AND SCATTERING IN QUASI-ONE-. . . 10 359

—2ik, I „—2ik,

Izi

rI 12+2&2

(28)

Appendix B shows T11 = 1 and T1z =0 near the third sub-band minima. By our mode rescaling arguments, all themode coupling constants will become zero at each newQlD subband minimum for the 5-function scatterer.

One can see this perfect transmission efFect even moreclearly by writing down the set of equations valid for onepropagating mode and one evanescent mode forE, &E &E, as

longer be a point in energy of perfect refiection in that re-gion because the bound state (which originally split off ofthe second subband) has dropped below the first subbandand therefore out of the scattering problem. We discussthe bound states of the 5-function potential in a quasi-one-dimensional wire in detail in Appendix A.

We are now interested in showing, from Eq. (28), howthe occupation probability of the evanescent mode accu-mulates as the Fermi energy approaches the second sub-band. The solutions to Eq. (28), valid in the energy rangefrom E1 & E & Ez, are

or, writing this slightly difFerently,

—2ik, (~, —c, )=r„c,+r„c,,—2~,C, =r„C,+r„C,. (29)

I » 2vz

I 22+ 2&2

T1 = 1+r

» 2kl

2K2

I 22+2 2

leading to the transmission coefficient T1, .2 2 —1

(30)

(31)

Including more evanescent modes in Eq. (28) will notaffect the structure of the equation, since their total effectis just to alter the mode coupling constants in a qualita-tively unimportant way. The first row of Eq. (28) or Eq.(29) asserts that the discontinuity in the derivative ofmode one at the defect is proportional to the probabilityamplitudes of each of the modes. The derivative jump ofthe evanescent mode, from the second row of Eq. (28),must also be supported by the buildup of the wave func-tion at the defect. The second row of Eq. (28) emphasizesthat the modes are coupled, since if the amplitude of theevanescent mode is taken to be zero, the propagatingmode must also have zero amplitude.

Setting ~2=0, meaning that there is no change in thederivative of mode two at the defect (mode two stays aconstant value for the entire length of the wire), givesperfect transmission of mode one, T» = 1 when E =Ez.To obtain this result from Eq. (28) we use the identityI z, I 1z=I »I zz. The analogous result holds for highersubbands; all the modes have perfect transmission when-ever the electron energy aligns with the bottom of a sub-band. Equation (28) asserts that, at the bottom of thesecond subband, the evanescent mode occupation willbuild up to exactly the right amount to compensate forthe derivative jump required of mode one at the scatterer.

There is another interesting feature of Eq. (28). If the5-function potential is attractive, that is if y (0, then it ispossible to have I 22+2~2=0. Or more generally, if thereare m total modes, then I zz 3 m+2~2=0. We show inAppendix A that the energy satisfying this equation cor-responds to a quasi-bound-state in the 5 function splittingoff from the second subband. In that case Eq. (28) pre-dicts zero transmission, that is T»=0. A qualitativelysimilar thing happens in higher subbands as we show inthe next section, although the transmission does not fallto zero for the higher subbands. So the presence ofevanescent modes in the wire leads to perfect transparen-cy when ~„=0for any n, and we can also lead to perfectopaqueness in the lowest subband if the 5-function poten-tial is negative but not too strong. If the 5 potential is sostrong and negative that I 22 3 +2~2=0 has no solu-tions in the energy range E1 & E &E„then there will no

Since there is only one propagating mode we have1 = T11 +R 11 ~ The wave-function amplitude in theevanescent mode is

Bzt1Z =

A1

I 21

I 22+2~2

= I 2, (I'22+ 2K2)— (32)

leading to a building pp of probability density

2Bz

A1'2

rl1 r22 (r22+2K2) + (2K2)» 2

2k,(33)

The transmission coefficient T1, from Eq. (31), as well

as the size of the evanescent mode ~t, 2~ =~82~ /~A, ~

from Eq. (33), are shown in Fig. 3. The strengths of the 5functions are given in the figure caption and otherrelevant parameters at the beginning of the next section.Figure 3(a) shows the case of a repulsive 5-functionscatterer. The two solid curves in Fig. 3(a) are for a weakrepulsive scatterer, while the two dotted curves describe aslightly stronger repulsive scatterer. The transmissioncoefficient T» is shown near the top of Fig. 3(a). Perfecttransmission, T» =1, occurs when the incident electronenergy aligns with the second subband minima atE =E2=25 meV. The transmission coefficient T» de-creases as the scatterer is made stronger. The electronprobability to occupy the evanescent mode, ~t, 2~' shownnear the bottom of Fig. 3(a), grows steadily as the energyapproaches the second subband. ~t, 2~ qualitatively fol-lows the density of evanescent states. As the strength ofthe repulsive scatterer increases and the transmissionprobability T» correspondingly decreases, the probabili-ty of occupying the evanescent mode increases. This is areasonable result since (1) a inore opaque barrier reducestransmission and (2) the evanescent modes must be popu-lated by scattering from the incident modes, and this

10 360 PHILIP F. BAGWELL

0.8-

~ ~~ggO~syIO ~

~~yyO ~JIay% ~++

ll

0

I

II

ll~ ~ 0 ~ ~ 0 ~I ~

l2

I

)t}~ f /20$$0~ 0

I I I

15 21 27 33Energy (meV)

have more than ten times the number of electrons storedin evanescent modes near the defect in Fig. 3(b) for theattractive scatterer as compared with the repulsivescatterer of Fig. 3(a). This is due to the quasi-bound-statenearby in energy. We shall see in the next section, how-ever, that if the scatterer is made so strong that thequasi-bound-state energy moves below even the first sub-band, the transmission coefficient Tii in Fig. 3(b} for theattractive scatterer will evolve to qualitatively resemblethe T„found in Fig. 3(a) for the repulsive scatterer. Forthe moderately strong attractive scatterer, evanescentmodes wi11 then become less important. Another impor-tant difference between the attractive and repulsivescatterers is that, for the attractive scatterer of Fig. 3(b),the occupation of evanescent mode two first rises, reachesa maximum near the quasi-bound-state energy, and thenfalls as the energy approaches the second subband mini-ma. This is in contrast to the repulsive scatterer of Fig.3(a) where the occupation of evanescent mode two in-creases continuously as the energy approaches the secondsubband minima.

IV. RESULTS AND DISCUSSIONOF 5-FUNCTION SCATTERER

.2-

0.0-I I I I

9 15 21 27 33Energy (meV}

FIG. 3. Transmission coef6cient T» through 5-functionscattering potential V(x,y)=y5(x)5(y —y, ) shown next to thestrength

~t » ~' of evanescent mode two. Perfect transmission

T» =1 results when E =E&=25 meV. (a) shows the case of arepulsive scatterer having strength y=10 feVcm (two solidlines) and y =SO feV cm' (two dotted lines). As the scatterer ismade stronger, transmission decreases and more electrons arestored in evanescent mode two. (b) shows an attractive scattererof strength y= —70 feVcm'. The transmission coeScient T»(solid line) becomes zero at the quasi-bound-state energy, whilethe number of electrons ~t» ~' (dotted line) stored in evanescentmode two reaches a maximum near the quasi-bound-state andthen decreases as the incident electron energy approaches thesecond subband minima.

scattering is enhanced if the scattering strength increases.Figure 3(b) shows T» (solid line) and ~ti2~ (dotted

line) for an attractive scatterer which is slightly weakerthan the repulsive ones of Fig. 3(a}. The point where

T» =0 is the quasi-bound-state energy of an electron in

the attractive scatterer which has split off from thesecond subband. Again T»=1 at the second subbandminima. The building up of the evanescent mode

~ t, 2 ~is

largest near the quasi-bound-state energy, but its max-imum does not occur at the bound-state energy. Equa-tion (33) for ~t&z~ has a Lorentzian-like shape whose

peak is above the bound-state energy. Note also that, forthe attractive scatter, the occupation of the evanescentmodes is much larger than for a repulsive scatter. We

In this section we graph the detailed solutions to theinfinite set of coupled Eqs. (4) for the 5-function scattererwhich are written down in Appendix B. We display allthe intersubband and intrasubband transmission andreflection coefficients for the single 5-function scatterer asfollows: As in Sec. III, we consider both a repulsive andan attractive scatterer. In Fig. 4 we consider a weakrepulsive 6-function scatterer, while Fig. 5 shows a weakattractive scatterer of approximately the same strength asthe repulsive scatterer. As the repulsive scatterer is madestronger, the energy dependence of the transmission orreflection coefficients does not qualitatively change. Foran attractive scatterer the energy dependence of thereAection and transmission coefficients does qualitativelychange as the scatterer is made stronger due to the move-ment of the quasi-bound-state. We do not show thetransmission coefficients for the case of a strong attrac-tive scatterer because they qualitatively resemble the onesfor the repulsive scatterer in Fig. 4. As the attractivescatterer is made stronger a11 the graphs of Fig. 5 wi11

evolve into ones qualitatively resembling Fig. 4. We thendisplay the two-probe and four-probe Landauer conduc-tance for these sets of scatterers in Figs. 6, 7, and 8.

For definiteness in all the calculations, we consider aninfinite square well along the y direction which we take tobe 8'=300 A wide. We take the mass of the electron tobe the effective mass for GaAs, that is 0.067 of the free-electron mass. For this wire, the subband energies areE& =6.236 meV, E2=24.94 meV, E3=56.12 meV, andE4=99.78 meV. We place the impurity —,', of the dis-

tance across the wire. We include 100 total modes in ourcalculations. These are the same parameters used in Fig.3. It is easy to include a greater or fewer number ofmodes, but including more modes does not have a greatqualitative inAuence on the results. As we argued in Sec.III, if we include enough modes such that, for the highest

EVANESCENT MODES AND SCATTERING IN QUASI-ONE-. . . 10 361

mode n we satisfy 2v„&)2mylWA', including more

evanescent modes makes only a small quantitativediff'erence in the results. Using Eq. (25) we see that, for arepulsive scatterer, including more evanescent modes in-

creases the transmission because the mode coupling con-stants are made smaller. In our rescaling procedure eachadditional evanescent mode then multiplies the couplingconstants by a number less than one. For an attractivescatterer where I

„„

is negative, including more evanes-

cent modes decreases the transmission because each addi-tional evanescent mode then multiplies all the couplingconstants by a number greater than one.

Figure 4 shows the transmission probabilities for arepulsive 5-function scatterer having y=10 feVcm . Orexpressed more usefully y/W =1.111 meV, so that thepotential is relatively weak. Consider first the intrasub-band transmission and reflection coefficients. Figure 4(a)shows that the transmission probability T, &

is unitywhenever the electron energy aligns with the bottom of a

new subband. Figure 4(b) shows a similar result for T2zand T33 The reflection coefficient R» is shown in Fig.4(c). Note that R» is zero at the minima of the secondand third subbands and undergoes no discontinuouschange as does T». Reflection into the lowest normalmode is suppressed at the bottom of the second and thirdsubband. A similar behavior for the intrasubbandreflection coefficient R22 is shown in Fig. 4(d). If only thelowest mode can propagate, the intrasubband transmis-sion and reflection coefficients T» and R» have the samefunctional form as those through a 5-function potential inone dimension given in Appendix A.

The intersubband transmission coeflicients T,z, T]3,and Tz3 are shown on Figs. 4(a} and 4(b) multiplied by afactor of 10 so they can be clearly seen. The intersub-band transmission and reflection coe%cients are equal forthe 5-function scatterer, that is R,&=T,z, etc. If therewere no scattering the intersubband transmission wouldbe zero, so the intersubband transmission should increase

1.0- (b)

22

4-

.2-

0

I

Iu 10 T(SIl~~ I ~~a~O ~~ ~ we~

~~I I I I

20 40 60 80 100Energy (meV)

4-

~2

0

L. 10T~@~~+~eaeaa~y+

I I I I

2Q 4Q 60 80 1QQ

Energy (me V)

1 0 (c}

.8- II—10 RII

22—100 R~~

4- 4-

0I I I I

20 40 60 80 100Energy (meV)

I I I I

0 20 40 60 80 100Energy (me V)

FIG. 4. Transmission coeScients for scattering from a weak repulsive 5-function potential in a quasi-one-dimensional wire having0

strength y=10 feVcm . We assume infinite square well confinement of width 8'=300 A. (a) shows T», T&p=T2l =Rl2=R2, , and

T$3 T3& =R»=R». (b) shows T», T33 and T»=T»=R»=R». (c) shows R» and (d) shows R» all as a function of the incidentelectron energy. The overall shapes of the reAection coeScients R„canbe understood from the golden-rule scattering rate as de-scribed in Appendix B. In addition, perfect transmission with no mode conversion occurs whenever the incident energy aligns with anew Q1D subband. The functional form of each curve is given in Appendix B.

10 362 PHILIP F. BAG%V.LL 41

22

~6-4-

~2-

I I I I

Q 2Q 40 60 80 100Energy (me V)

0-I

0 20I I

40 60Energy (meV}

I

80 100

1 0 (c) (d)22—10 122

0.6-4-

~2 .2-

~0-I

0 20 40 60Energy (meV)

I

80 100

.0-0 20

I I

40 60 80 100Energy (me V}

(e)

l2

I3

25

I

0 20

rI I I

40 60 80 100Energy (meV}

I

0 20I

40I

60Energy (meV}

I

80 100

FIG. 5. Scattering from a weak attractive 5 function having strength y= —6 feV cm'. (a) shows T», (b) gives T» and T33 (c)shows R», (d) shows R», (e) gives T», while (f) shows T&3 and T». The qualitatively diferent behavior of these transmissioncoefficients from the ones in Fig. 4 arises from the presence of new quasi-bound-states forming in the attractive 5-function potential.For example, the new dips in the intrasubband transmission T&&, Tz2, and T» occur well before reaching subbands 2, 3, and 4, the dis-tance between these dips and the subband minima being simply the bound-state energy as we show in Appendix B. The intrasubbandrefiection increases strongly at the quasi-bound-state energy as shown in (c) and (d). The intersubband transmission and reflection,given in (e) and (f), also rises strongly at the quasi-bound-state.


with increasing strength of the scatterer. This is indeedtrue, although not shown in the figures. In Fig. 4 the in-tersubband transmission is small because the potential isrelatively weak. At the onset of the second subband in

Fig. 4(a} only about 6% of the incident carriers are con-verted into the second normal mode through T,2, and4—5% are converted into the third normal mode via T,3

at the bottom of the third subband. Figure 4(b) givesonly between 1%%A and 2%%uo conversion from the second tothe third mode at the bottom of the third subband viaT23 ~

We can understand some features of Fig. 4 by arguingfrom the Fermi "golden-rule" scattering rate. To do thiswe do not consider the intrasubband transmission T&&,

T~q, or T33 as they are simply the result of leftover parti-cles which did not scatter and can be obtained from therequirements of current conservation. Consider first theintersubband transmission T&2, T&3, and T23. The inter-subband transmission has a maximum near the onset of asubband and decays like the inverse square root of energyaway from the maximum. This can be understood from aFermi's "golden-rule" viewpoint, where the probability ofscattering is proportional to the final density of states inthe subband which decays like I/~E. In Appendix B

we show that the dominant term in the intersubbandscattering probability is indeed given by an expressionsimilar to the golden rule. The intersubband transmis-sion and reffection coefficient T,2=R, z in Fig. 4(a) alsosho~s interesting behavior around the bottom of thethird subband, staying zero on both sides of the subbandminima. There is no scattering out of mode one intomode two at the bottom of the third subband. We haveyet to find a good explanation for this lack of mode con-version or reflection at the subband minima. However,the overall shapes of the transmission and reflectioncoefficients are still well understood by golden-rule argu-ments.

Given the golden-rule-like shapes of the intersubbandtransmission and reflection coefficients and the intrasub-band reflection, we can argue for the shape of the in-trasubband transmission. Let us do so for T». Becauseparticles must be conserved so that 1 = T» + T,2

+R,2+R», and since R» =0 on both sides of the sub-band minima, the drop in T» after reaching perfecttransmission at the second subband must he equal toT,z+R» =2T&2, or just twice the intersubband transmis-sion coefficient. This is shown in Fig. 4(a). Similarly, thediscontinuity in T» in Fig. 4(a} at the minima of thethird subband is just twice T,3.

Next, let us examine the scattering coefficients for anattractive potential. Figure 5 shows a 5-function scatter-er of comparable strength to the one in Fig. 4, but when

I I I I

0 20 40 60 80 100Energy {meY}

FIG. 6. Two-probe conductance through a 5-function defectin the quasi-one-dimensional wire in units of 2e /h. The solidline corresponds to the repulsive scatterer from Fig. 4, while thedashed line gives the conductance of the attractive scattererfrom Fig. S. When the electron energy aligns with a subbandminimum, the conductance through the defect is equal to theballistic conductance. At these special energies the wire is per-fectly transparent as if no scatterer were present. There is onlya small difference between the conductance for the weak repul-sive scatterer and the ideal ballistic conductance throughout theentire range of electron energies. For the attractive scatterer,the new dips in conductance correspond to quasi-bound-statesdeveloping in the wire. The distance in energy from these dipsto the subband minimum is the quasi-bound-state energy. Notealso that, even though the repulsive scatterer is stronger, theconductance of the attractive scatterer is much smaller due tothe presence of the quasi-bound-state.

I2.0-

Cd

V

o 1.0-V

0 20I I

40 60 80 100Energy {meY}

FIG. 7. Two-probe conductance in units of 2e /h for an at-tractive scatterer having y = —8 feV cm (solid line), y = —9feV cm (dotted line), and y = —20 feV cm (dashed line). Begin-ning with the dotted line from Fig. 6 showing the weakest at-tractive scatterer having y= —6 feVcrn, the overall conduc-tance level decreases and the new dips corresponding to thequasi-bound-states move lower in energy as the scatterer ismade more attractive. As the scatterer becomes so attractivethat the quasi-bound-states move below the bottom of the nextlowest subband, the new dips first disappear and the conduc-tance then increases as the scatterer is made stronger. Thisunusual effect occurs because the bound states have now movedbelow the energy range in which they can block conduction.

10 364 PHILIP F. BAGWELL 41

the scatter is made attractive. The strength of thescatterer in Fig. 5 is y= —6 feVcm, or equivalentlyy/W = —0.666 meV, 60% of the strength of Fig. 4.The transmission coeScient T» through this potential isshown in Fig. 5(a). The new dips in T» immediately be-fore each subband minimum correspond to a quasi-bound-state forming in the wire. The distance in energybetween each local minimum in T, ] and the followingsubband is simply the quasi-bound-state energy. Al-though these new minima appear to be immediatelybelow the subband minima, they are in fact at a lower en-ergy. We can see this clearly from Fig. 3(b) where the at-tractive scatterer is stronger. Since the scatterer in Fig. 5is very weak, the quasi-bound-states lie very near in ener-

gy to their respective subbands. Figure 5(b) shows T22and T33 for this potential. %e see again that intrasub-band transmission decreases at the quasi-bound-state en-

ergy and is perfect at the subband minima.The intrasubband reflection R&& and R22 increase at

the quasi-bound-state energies and are zero at each suc-cessive subband minimum as shown in Figs. 5(c) and 5(d),respectively. Intersubband transmission also increases atthe quasi-bound-state energies as shown for T,2 in Fig.5(e) and again for T» and T23 in Fig. 5(f). Note thatboth the intrasubband reflection and intersubbandtransmission are much larger for this attractive scattererthan for the repulsive scatterer we considered in Fig. 4,even though the scattering strength is weaker than in Fig.4. This is also due to the quasi-bound-state nearby in en-

ergy.Now we turn to a study of the Landauer conductance

through these 5-function scatterers. Figure 6 shows thetwo-probe conductance from Eq. (15) for both the repul-sive 5-function defect of Fig. 4 (solid line) and the attrac-tive 5-function defect of Fig. 5 (dashed line). The con-ductance for the weak repulsive scatterer is nearly thesame as that of a perfect ballistic wire throughout thewhole range of Fermi energy, and is exactly equal to theballistic conductance when the Fermi energy aligns witheach new subband minima. The shoulders of the quan-tized ballistic conductance steps are slightly rounded dueto increased reflection immediately above each subbandminima. The weak attractive scatterer, however, has amuch smaller conductance reflecting the presence of thequasi-bound-state nearby in energy. The extra dips in theconductance through the attractive 6 function againoccur before reaching each subband, the difference beingjust the quasi-bound-state energy. The conductance ofthe attractive 5 function is also exactly equal to the ballis-tic conductance when the Fermi energy aligns with newsubband minima.

Figure 7 shows the two-probe conductance evolving aswe increase the strength of the attractive scatterer fromy= —6 feVcm, shown as the dashed line in Fig. 6, toy= —8 feVcm (solid line), y= —9 feVcm (dotted line),and y = —20 feV cm (dashed line). Note that, as for therepulsive scatterer, perfect transmission still occurs at thesubband minima. However, the additional quasi-bound-state in the attractive scatterer causes increased reflectionwell before the subband minima are reached, leading toan additional dip in the conductance (solid curve). As the

2.Q

o 1.2-.ICCg

~- Q.8 - 'tOIK

Two Probe—Four Probe

0.4-

oo-~=—I I I

0 20 40 60Energy (me V)

80 100

3.0- Two Probe—Four Probe

Cg 2.0-

1.0- (b)

I I I I

0 20 40 60 80 100Energy {meV)

FIG. 8. Two-probe resistance (solid line) and four-proberesistance (dashed line) in units of h/2e for the case of (a) therepulsive scatterer from Fig. 4 and (b) the attractive scattererfrom Fig. 5. The overall resistance is much lower for the four-probe measurement due to the absence of contact resistance.The four-probe resistance Rf p 1, is zero at each subbandminimum in both (a) and (b) due to the perfect transmission.Rf p 1, rises just above each subband minimum to equal thetwo-probe resistance as a result of the screening described inRef. 15 for both the attractive and repulsive defects. In addi-tion, for the attractive scatterer of (b), R f p Q also rises im-mediately below each subband minimum due to increasedreAection at the quasi-bound-states.

strength of the attractive potential is increased, the con-ductance becomes smaller and the dip moves lower in en-ergy until the quasi-bound-state drops below the nextsubband as shown by the dotted curve, where the conduc-tance resembles a "cobweb" (dotted curve). As the po-tential is made even more attractive (dashed curve), theconductance increases and begins to resemble the con-ductance through a repulsive 5 function. This is becauseeach quasi-bound-state energy which originally split offfrom, say, subband i, has now moved below the bottom ofthe next lower subband i —1, and therefore out of theproblem. It is paradoxical that, as the scatterer is madestronger over this small range of strengths, the conduc-tance actually increases. The bound states have nowmoved out of the way and the electron can transmit


through the defect. Usually stronger scatterers lead to areduced conductance. But we see this is not necessarilythe case when the electrons are confined to move in awire.

Figure 8 compares the two- and four-probe Landauerconductance for the scatterers from Figs. 4 and 5. Sincethe four-probe conductance is so large, we plot its in-verse. Figure 8(a} shows the resistance of the repulsivescatterer having the same strength as in Figs. 4 and 6.The solid line in Fig. 8(a} is the two-probe resistance, theinverse of the two-probe conductance from Fig. 6. Thedashed line in Fig. 8(a) shows the four-probe resistancefor the same repulsive scatterer. The four-probe resis-tance is of course much smaller than the correspondingtwo-probe resistance due to the absence of any contactresistance. The four-probe resistance is close to zero ex-cept when the electron energy is slightly above each sub-band minimum, where it suddenly rises to equal the two-probe resistance and then falls again. The four-proberesistance being zero slightly below each subband is aconsequence of the scatterer being perfectly transmitting,while the rise in the four-probe resistance slightly aboveeach subband minimum is a consequence of screening. '

Near the bottom of each subband the voltage divisionfactor in Eq. (15) is dominated by the transmitted orreflected waves for that subband. Since each intrasub-band reflection coefficient is one at the bottom of itsrespective subband, while the intersubband reflection isinitially zero, the voltage division factor is simply 1 andthe two- and four-probe conductances are equal at thesespecial points.

Figure 8(b) shows the resistance of the attractivescatterer having the same strength as in Figs. 5 and 6.The two-probe resistance (solid line) is the inverse of thedashed line from Fig. 6. The four-probe resistance(dashed line) for the same attractive scatterer is againlower than the two-probe resistance due to the absence ofcontact resistance. Both resistances increase sharply atthe quasi-bound-state energies, and are both equal andinfinite at the quasi-bound-state in the first subband dueto perfect reflection at that point. At subsequent quasi-bound-states, the four-probe resistance is smaller becausethe reflection at the quasi-bound-states in higher sub-bands is not perfect. The four-probe resistance is againzero at each subband minimum, and again rises to equalthe two-probe resistance just above each subband. Nearthe subband minima the resistance for the attractivescatterer first rises due to the presence of the quasi-bound-state, then falls to zero due to the unexplainedmode "decoupling, " then rises again to equal the two-probe resistance as a consequence of screening.

We have also numerically examined the transmissionthrough two 5-function scatterers in a narrow wire, andfind unusual modifications to resonant tunneling whenev-er the Fermi energy is near the second subband minima.In particular resonant transmission is suppressed whendistance between the two barriers is equal to an odd num-ber of half wavelengths of the lowest mode, butunafFected when the distance between the barriers equalsan even number of half wavelengths. It is also possible toexactly solve the Dyson equation for scattering from a 5-

function defect in a QID wire, including mode conver-sion and coupling to the evanescent modes. The sametransmission coefficients are obtained as in Appendix B.

k„= (E E„)— (34)

such that the c„(x)'sin the region 0(x (L can be ex-

pressed

c„(x)=C„e" +D„eif n denotes a propagating mode in region II and

(35)

c„(x)=C„e" +D„e" (36}

if n denotes an evanescent mode in region II. Since re-gion III has the same confinement potential as region I,the normal modes y„(y)and wave vectors k in region IIIwill be the same as for region I. We take the coefficientsfor x motion to be E„andF„in region III where

ik„(x—L) —ik„(x—L)(37)

describes the propagating modes for x & L.Following our discussion of the scattering boundary

conditions in Sec. II, enforcing continuity of the wavefunction and its derivative at each boundary leads to thetransfer matrix

k1+

a kp

k+ —,

' g f dy Y& (y)y (y)a kp

Cp=-,'y f d3 Ypy}x.(y}

kf dy Y& (y)y (y) 1 — Aa k(

k+ —,

' g f dy Y&(y)y (y) 1+ Ba kp

(38)

(39)

V. FINITE-RANGE SCATTERER

In Sec. III and IV we considered the transmission andreflection from a single 5-function scatterer inside a nar-row wire. In this section we examine the next simplestcase where the scatterer inside the Q1D wire is a rectan-gle with a finite width and length. Recall the geometry ofFig. 1, shown there for an arbitrarily shaped defect.Analyzing the transmission through a rectangular defectwill allow us to understand how to calculate transmissionthrough an arbitrarily shaped defect.

As in Sec. II, the wave functions c„(x)in region I havecoefficients A„and B„givenby Eqs. (7) and (9), wave

vectors k„given by Eq. (5), and are described by the nor-mal modes g„(y). In region II we will also havecoefficients parametrizing the exponential solutions for xmotion C„andD„.But now, due to the presence of theextra defect, we will have a new set of subbands E„andanew set of normal modes Y„(y)which must be calculatednumerically by diagonalizing a matrix expanded in theold basis set y„(y). In addition, we have a new set ofwave vectors k„defined by


describing the mode coupling at x =0, the transfer ma-

trix describing the propagation on top of the rectangularbarrier as

ikpL (—ikpLCp=Cpe, Dp=Dpe (40)

and which does not couple the modes, and finally thetransfer matrix

f 4' x (3') ~p(P) 1+p

+-,' y f d3 x,'(x) 1'p(3»p

F, =-,' X f d3 Xr'(~)&p() )

p

+-,'r f d3 x,"(3)&pu)p

kpCp

r

kpp '

r

kpCp

r

rp~+ Dl3r.

(41)

(42)

incorporating mode mixing at the x =L potential step.Multiplying these three transfer matrices numerically

will give all the transmission coefficients through the rec-tangular defect. However, there is a numerical problemwith the growing real exponential for D„in Eq. (40) if weinclude too many evanescent modes, or if the length ofthe scatterer L is made too long. The highest evanescentmodes, which should become unimportant to the physicsof the problem, wind up dominating the transmissionproperties as well as violating particle number conserva-tion. In place of multiplying the transfer matrices, if weinstead cascade scatter matrices as described in Ref. 2 weavoid these numerical problems. An arbitrarily shapeddefect can be broken up into many rectangular shapeddefects of short lengths, so that equations similar to Eqs.(38), (39), and (40) can be used to calculate the transmis-sion and reflection coefficients through an arbitrarilyshaped defect in a narrow wire.

We display the two-probe Landauer conductance for arectangular shaped scatterer in Fig. 9. Figure 9(a) showsa repulsive scatterer that is 50 nm long (along the x direc-tion), 8.7 nm wide, offset 1.2 nm from the side of the wire,and 50 meV tall. We include six total modes in our cal-culations, and again assume the original confinement po-tential to be an infinite square well of 30 nm width. Theshape of this defect potential forms an extra constrictioninside the Q1D wire. Extra oscillations analogous to theRarnsauer resonances appear in the conductance of Fig.9(a), known from many previous calculations includingRefs. 9, 11, and 12. These Ramsauer-like resonances ap-pear in all the inter- and intrasubband transmission andreflection coe%cients, though we do not show these here.We emphasize, however, that the rises in conductancecorresponding to the "quantized steps" do not occur atthe original subband energies E„of6.23, 24.9, 56.1, and99.7 rneV, but rather at the new subband energies E„of10.3, 39.4, and 78.3 meV. Due to the presence of evanes-cent modes below the E„,some carriers can leak throughbelow the new subband minima as shown in the figure.Also, nothing unusual happens when the Fermi energy

30 - (n)

Ir 2.0-VC5

V

g 10-V

I

0 20I

40I I

60 80 100Energy (me V)

3 0 - (b)

I2.0-V

CO

V'0C 1.0-V

0.0-0

I I I I

20 40 60 80 100Energy (meV)

FIG. 9. Conductance in units of 2e'/h through a finite sizerectangular barrier whose dimensions are given in the text. (a)shows a repulsive barrier while (b) considers the same size bar-rier when the sign of the potential is reversed. Ramsauer-likeresonances are observed in (a) due to the finite barrier size. The"quantized steps" in (a) occur at the new subband minima insidethe barrier, not those of the clean wire. Multiple quasi-bound-states where the conductance falls sharply occur in (b) again dueto the finite size of the defect. Transmission is no longer perfectat each minimum as was the case for the 5-function scatteringdefect.

rises above the 50-rneV barrier height as shown in Fig.9(a).

When we reverse the sign of the scatterer, making it at-tractive but leaving all other parameters unchanged, weobtain the two-probe conductance shown in Fig. 9(b).The new subband energies E„for those potential are at—20.6, 13.3, 44.0, and 84.3 rneV. Three bound states canbe seen in the conductance of Fig. 9(b) immediately be-fore the original second and third subband minima. Ourpoint in Figs. 9(a) and 9(b) is that, as we give the scatterera finite size, (1) we obtain geometrical resonances with thedefect analogous to the Ramsauer resonances, (2) therecan be multiple bound states if the scatterer is attractive,(3) there is more than one set of subbands in the problemwhich are important, and (4) the transmission is nolonger perfect at a subband minimum although this is not

41 EVANESCENT MODES AND SCA I I'ERING IN QUASI-ONE-. . . 10 367

shown in the figures. For example, T&& has quite a com-

plex behavior for both potentials considered in this sec-tion. Also we have verified that, as we shrink the size ofthe scatterer while maintaining the total area underneaththe potential, the results for transmission coefficientsevolve into those for the 5-function scatterer studied inthe preceding two sections.

VI. CONCLUSIONS

In this paper we have considered the scattering fromdefects in a confined geometry, in our case for electronsconfined to a quasi-one-dimensional wire. The scatteringproperties of electrons in these confined geometries arequalitatively different from the usual case of scattering in

open geometries due to the building up and storage ofelectrons in evanescent waves near the scattering defect.This phenomenon is well known for the analogous case ofelectromagnetic wave scattering in microwavewaveguides. We obtained both the intersubband and in-trasubband transmission and reflection coefficients byhand for the special case of a 5-function scatterer, andnumerically found these same coefficients for the casewhen the scatterer has a finite size. We then used thesecoefficients to study the two- and four-probe Landauerconductance through the defect.

For the case of the 5-function scatterer we found that,for electron energies equal to the subband minima, all thenormal modes completely decouple and that conductancethrough the defect is equivalent to the ballistic conduc-tance in agreement with Ref. 23. This unusual result isindependent of where the scatterer is located in the wire,the shape of the confining potential, and the subband sep-aration (i.e., independent of the incident electron wave-length). This "perfect transmission" effect thereforeseems to be independent of any exact geometrical featurein the problem except for the existence of confinement.These facts support our conclusion that the existence ofevanescent modes is the qualitatively new and importantfeature responsible for the unusual scattering propertiesin the confined geometry, rather than a geometrical reso-nance between the scatterer and the walls of the wire. Inaddition, if the 5-function scatterer is made attractive, aquasi-bound-state splitting off of each Q1D subbandforms in the potential and dominates the scattering prop-erties when the electron energy is near the quasi-bound-state energy.

For the finite size scatter which we explored briefly inSec. V, the overall features in the transmission ampli-tudes and the conductance qualitatively resemble those ofthe 5-function scatterer. However, the transmission is nolonger perfect at each subband minimum. %'e thereforeattribute the perfect transmission seen for the 5-functionpotential to the special shape of the scatterer, and con-clude that it is not a general feature for any possibleshape of the scattering potential. Reference 23 concludesthat any s-wave scatterer wi11 exhibit the perfecttransmission property. The 5 function is an s-wavescatterer while the square barrier is not. We can con-clude in general that the transmission is enhanced near asubband minimum and reflection increased near anyquasi-bound-states in the wire. The finite size scatter also

APPENDIX A: QUASI-BOUND-STATESFOR AN ATTRACTIVE 5-FUNCTION POTENTIAL

IN A QID WIRE

The bound-state energy and transmission coefficientthrough a 5-function potential V(x)= VO5(x) in one di-

mension can easily be evaluated to find

IE=—2@i

(Al)L

2

T(E)= 1+4k

where I =2m Vo/fi and the bound-state energy is a solu-

tion of the equation 2~+I =0 where K is the inverse de-

cay length.In this appendix we wish to discuss the quasi-bound-

states for a 5 function in a QlD wire in order to under-stand the similarities and differences between the Q1Dcase and the 1D case worked out in Ref. 30. The poten-tial V(x,y)=y5(x)5(y —

y, ) is attractive so that y &0.We consider a wave function which consists of a sum ofexponentially decaying modes as

I

x.(y»n

(A2)

so that the matrix equation for the B„'s,taking fourevanescent modes for definiteness, is

I ) )+2~, I &4 B,I »+2~, Iz3

I 33+2~3

I 43

I 24 B2

r~+ 2&4 B4

(A4)

The same method of rescaling the mode coupling con-stants applies for this case as in the previous discussion ofSec. III, so that for m total modes the bound-states ener-

gy is a solution of

exhibits geometrical resonances (analogous to the Ram-sauer resonances) in all the transmission coefficients, bothintersubband and intrasubband. A finite size scattererwhich is attractive can have multiple quasi-bound-statessplitting off of the higher confinement subbands, and theeffects of these can be seen in the transmission probabilitythrough the defect.

Note added in proof. We have learned after submittingthis manuscript of a complementary study of impuritiesin a constriction by Tekman and Ciraci, ' of a quantumcavity resonator by Peeters, and of the inhuence ofevanescent waves on electron localization in the limit of alarge number of impurities by Cahay et al.

ACKNOWLEDGMENTS

We thank Terry P. Orlando, Arvind Kumar, Mare A.Kastner, Charles Kane, Dimitri A. Antoniadis, and Hen-

ry I. Smith for useful discussions. This work was spon-sored by the U.S. Air Force Office of Scientific Researchunder Grant No. AFOSR-88-0304 and the U.S. Joint Ser-vices Electronics Program Contract No. DAAL03-89-C-0001.


11 2 +2K1 =0 (A5)

Call the energy which solves this equation E,b. Thisbound-state energy will satisfy E,b &E„sothat it isbelow the first subband energy. The energy E,b is a truebound state which is a stationary solution of theSchrodinger equation. However, if mode one is a propa-gating mode and we set k, =is, in Eq. (A4), we cannotobtain a solution to the equation. This implies there areno "truly bound" states in the wire which are degeneratein energy with any of the propagating modes. But if wethrow away all of the mode coupling to the propagatingmodes, in this case by neglecting the first row and firstcolumn of the matrix in Eq. (A4), we obtain a quasi-bound-state energy E2b below the second subband givenby

I 22 3 +2 =0where the quasi-bound-state solution E» to this equationsatisfies E2b &E2. If the 5 function is made attractiveenough we can even have E2b E1. The energies E2bE3b etc. , are important in understanding the scatteringproperties of electrons from the 5-function defect. Sincethe quasi-bound-states at energies Eb2, E», etc., actuallydo couple to the propagating modes of the wire, they willdecay with time. Only the bound state splitting off of thelowest subband at energy Eb, is a truly bound state.

Equation (A5} can be rewritten asm

1+ g =0. (A7)2Ki

(A6)

The bare mode coupling constants appear in Eq. (A7),not rescaled by any of the evanescent modes. ,Equation(A6) can be rewritten similarly as

m

1++ "=0,i=2 2K

(A8)

and so on for all the possible quasi-bound-states. Clearly,the terms in the sum become smaller and smaller quiterapidly as the ~ s become larger and larger.

Sketching the terms in the sums of Eqs. (A7) and (A8)shows that there can only be one bound-state or quasi-bound-state solution to each equation. The sketch alsoshows that the presence of all the other evanescent modesacts to lower the bound-state energy below its value inthe absence of all the other evanescent modes. Thereforean upper bound on the solution of Eq. (A7), for example,is I 11+2K1=0. Since the solution to this equation isbelow E„andthe solution to Eq. (A7) will have an evenlower energy as we can see graphically, we must have thebound-state solution to Eq. (A5) satisfy E,b & E, .

Next consider Eq. (A8), which describes only modestwo and higher allowed as evanescent waves. The quasi-bound-state energy solution to Eq. (A8) does not have tobe below the lowest subband, in contrast to the case ofthe 5 function in 1D. Even if the first mode is propaga-ting, there can still be a quasi-bound-state split of fromthe second subband given by the solution to Eq. (A8) atany energy below E2. An upper limit to the energy Ebzis found by solving I ~2+ 2]&2=0.

APPENDIX 8: SOX UTIONSTO COUPLED SCATTERING EQUATIONS

In Eq. (Bl) g' denotes a sum over all the evanescentmodes, gi' denotes a sum over the propagating modes,and modes a and b are assumed propagating. Equation(81) also holds for the intersubband transmission ampli-tudes since F,„(E)=t,b (E) for ab The intr. asubbandtransmission amplitude t„(E)can then be obtained fromthe constraint of current conservation. The simple ana-lytic result of Eq. (Bl) can be interpreted in terms of theFermi golden-rule scattering rate as we argue below, andhas been obtained more transparently from the solutionof Dyson's equation for this same scattering problem.Equation (81) is a compact way of writing down the re-sults listed in this appendix, and can be obtained fromthem by the same type of algebra used to go, for example,from Eq. (A5) to Eq. (A7).

We now write down the solutions for the transmissionamplitudes and transmission probabilities for m totalmodes as the number of propagating modes ranges be-tween one and three. When transport is only in thelowest subband, namely, E, &E &E2, we must solve thematrix Eq. (28) or Eq. (29) given in Sec. III to obtain

t„=1/d, = 1+r„,k1

t t*=~ 11 k 11 11 D1 1

where the denominator amplitude d, is

d1=1— 11,2 —m

2ik,yielding the denominator

(83)

(84)

D, =d, d', =1+ (85)

Here we wish to write down the general solution to theinfinite set of coupled scattering equations, Eq. (4},for thecase of a single 5-function scatterer. For this case Eq. (4)can be expressed in a form similar to Eq. (23) for anynumber of propagating incident modes. We proceed asfollows: We write down the solutions first for the case ofa single propagating mode, then for two propagatingmodes, then for three, etc. It is these solutions which wehave plotted in Sec. IV. In writing down these solutionswe will recognize a pattern that, when generalized, leadsto the transmission coefficients for any arbitrary numberof propagating modes. We have already considered theeffect of the evanescent modes in Sec. III, so our solutionsare valid for any arbitrary number of evanescent modesby simply rescaling the mode coupling constants. Thisrescaling procedure is only valid for the case of a single5-function scatterer in a wire.

Our main result in this appendix is to show that thereflection amplitudes F,b can be written as

~ab

2+k, kb(81)

1+ g (I'„„/2it„)+i g (I'„„/2k„)


Note that Eq. (83) has the same form as the transmissioncoefficient through a 5 function in one dimension given in

Appendix A, except that we make the substitution

r», z-The reflection coefficients are

2 22 trt ~ iz

2

=N, (E) —( l~y5(y —y;)~1) N, (E) (89)

r», 2—

2ik1

d 1

(86)

(87)2

r», z-ii k li il

1 1 1

from which the current conservation identity

R»+T» =1 (88)

is satisfied. Note also that 0& T» 1 since 1(D,so that the normalized two-probe conductance in units of2e lh is between zero and one when only the lowest sub-band is occupied.

We would like to give Eq. (83) for T» and Eq. (87) forR» an interpretation in terms of the Fermi golden rulefor scattering rates. Our viewpoint is that T» does notdescribe any scattering processes (for this problem), andis just the remaining particle flux left over by currentcontinuity after any scattering processes (reliection or in-tersubband transmission) subtract particles from the in-cident beam. The only scattering process when one sub-band is occupied is intrasubband reflection. The numera-tor for R» can be approximated as

I

which is reminiscent of the Fermi golden rule for transi-tion rates. The numerator of R» is simply the density ofinitial states in subband one, the square of a matrix ele-ment coupling subband one to itself through the scatter-ing potential, and the density of final states in subbandone. Each term in Eq. (Bl) can be similarly interpreted.

To lowest order in the scattering potential we wouldjust have T„=1 (i.e., the particle would pass through theobstacle undisturbed) and R» just equal to its numeratorfrom Eq. (89), which is the Fermi golden-rule transitionrate. But then particle continuity would be violated.Hence the denominator Di from Eq. (85) is determinedby the sum of the scattering probabilities for each processto the lowest order in the scattering potential. Thisdenominator D1 preserves particle continuity.

Now we consider transport for energies in which boththe first and second subbands are occupied. ForEz&E(E, we have two propagating modes and fourtransmission coefficients. Consider first the case whereonly mode one is incident on the scatterer. We obtain amatrix analogous to Eq. (23)

—2ik, r» —2ik,

rz1

r41

r1zI —2ik

r3z

r4z

r13

rz3

r33+2v3

r14

rz4

r34

I 44+2@4

&12

(810)

Solving the tnatrix Eq. (810) we obtain the transmissionamplitude t» as

r

k,R 11

— P11r 111

», 3—m 1

2k1 Dz(816)

rzz, 3-t = 1—

2iki=1+r11 The remaining intersubband transmission amplitudes and

transmission coefficients from Eq. (810) are

k, 1+1

rZZ, 3-m

2kz

where the denominator amplitude d z is

yielding the transmission coefficient T11'2

1

Dz(812)

rz1, 3 — 1

2'k d

k2 ~ r» 3 —m r223 —m 1T12 k 1212 2k 2k D '121 1 2 2

(817)

(818)

d =1- r», 3—m2

rzz, 3-2ikz

(813)Equation (23) describes the transmission properties fortwo occupied subbands when mode two is incident on thedefect. We obtain the solutions to Eq. (23) as

and the denominator Dz is

Dz —dzd2 —1+ r„, r„,2k, 2kz

(814)

Note that from Eq. (812) we can recover Eq. (83) by set-ting kz~i~z. The reflection coeScient R» is obtainedfrom

r1Z, 3-m=~21

2lk1 dz(819)

so that Tz, =T,z and

r11,3— r22, 3 —mt~, t2, — —R2, (820)

r11,3—m 111 (815)

r11,3—m1— =1+rzz,2

(821)


kzTzz — tzz t 22

— 1+2

2ikz d

I ]],3—m

2k,1

Dz(822)

(823)Iii4 —m 1R»—

(832}

(833)

kzRzz =

k rzzrzz =2

2I 223 m 1

2kz Dz

The current conservation identities

T»+ Tiz+R»+R (2=1,Tz]+ Tzz+Rzi+Rzz —1

are also satisfied. The identity

1T» +Tiz+ Tzi + Tzz —1+

Dz

(824)

(825)

(826)

(827)

The remaining coefficients are

~21,4—m 1

27kz d3

k2 I 11,4—m I 22, 4—m 1

D1 1 2 3

I3i4—m 113 2 k d l3

3 Pi),4 m 33,4 m 1

k 2k 2k D1 1 3 3

(834)

(835)

(836)

(837)

limits the normalized two-probe conductance to be be-tween one and two for two occupied subbands since1&D2

We would like to consider a Fermi golden-rule-like in-terpretation of the transmission and reflection coefficientsfor two occupied subbands. Analogously to our interpre-tation for R» when one subband is occupied, given byEq. (89), the numerators of R

&&from Eq. (816), T&2 from

Eq. (818), R, 2 from Eq. (818), T2, from Eq. (820), R2,from Eq. (820), and R22 from Eq. (824) can be regardedas the golden-rule scattering rates for their respectivescattering processes. The denominator Dz from Eq.(814) can similarly be regarded as the sum of all thescattering processes.

For transport for three occupied subbands in the ener-

gy range E3 &E & E4, the pattern from our calculation ofthe transmission coefficients for one and two occupiedsubbands continues as before. The matrix equations wesolve to find all the scattering coefficients are very similarto Eq. (23) from the text and Eq. (810) from this appen-dix, so we do not write them down. Instead, we merelycatalogue the solutions for three propagating subbands.The transmission amplitude t» is

~11,4—m

22 1 —2k

I 33,4-m

2ik3

1 —1+rzz,3

kzT22 t22t 22 1 +

k2

r„, r„,2k, 2k,

I 22, 4 — 1

d 3

1 22, 4—R zz

— rzzr zk2 2k2

I 32,4-3 d 3

1

D3

I &24—m 1

22k, d

k, I„ IT21 k 21 21 2k 2k D 21

2 1 2 3

giving Tz& =T,z. We write

(838)

(839)

(840)

1

D3

(841)

(842)

(843)

(844)

I 22, 4—1—2ik2

I 33,4—m

2ik3

1 =1+r»,d3

(828)k3 P33 4 Pzz 4T = t t*=

D2 3 2 3(845)

k, r„,. r„,.1+ ' +1 2 3

where the denominator amplitude d3 is

'2

D3

(829)

I i34—m 1

2k, d, "''k I I ii4T = i t'=

31 k 31 31 2k 2k D 313 3 1 3

(846)

(847}

and

d =1—~11,4—m

2ik,

I 22, 4—

2ikz

I 33,4—m

2ik3(830)

so that Ti3 T3],

I 23,4-m

27 2 3(848}

'r r r2k

&2kz 2k3

The reflection coefficient is obtained from(831}

k2 P33 4—m P224 —m 1T = t32 k 32 32 2I 2k D 32 7

3 3 2 3

leaving Tz3 = T3z, and finally

(849)

41 EVANESCENT MODES AND SCATT'ERING IN QUASI-ONE-. . . 10 371

~22,4-m33

~33,4—m

2ik3

1 = 1+r33,3

(850)T3] + T32 + T33 +R 3] +R 32 +R 33 1

are also satisfied. The identity

(856)

k333 33t33 1+

k3

~33,4-m

2ik, d, '

11,4—+ 22,4—

2k( 2k2

1

D3

(851)

(852)

T„+T)2+ T)3+ T2] + T22+ T23

1+ T3) +T32+ T33 2+3

(857)

k3 I 334

k 2k3 3

1

D3

The current conservation identities

T))+T)2+ T)3+R ))+R)2+R )3—1,

T2i + T22+ T23+R2) +R22+R23 =1,

(853)

(854)

(855)

limits the two-probe conductance to be between two andthree when three subbands are occupied since1 KD

The equations in this appendix are plotted in Sec. IV.By continuing the pattern from the above equations, wecan obtain a general solution for any number of propaga-ting modes in the waveguide scattering from the 5-function defect given compactly in Eq. (81).

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evanescent modes and scattering in quasi-one-dimensional wires

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