Numerical analysis of ballistic-electron transport in...

PHYSICAL REVIEW B VOLUME 52, NUMBER 11 15 SEPTEMBER 1995-I

Numerical analysis of ballistic-electron transport in magnetic fieldsby using a quantum point contact and a quantum wire

T. Usuki, * M. Saito, M. Takatsu, R. A. Kiehl, and N. YokoyamaQuantum Electron Deuices Laboratory, Fujitsu Laboratories Ltd. , 10-1 Morinosato Wa-kamiya, Atsugi 243-01, Japan

(Received 10 January 1995)

We report the numerical analysis of our experimental results for electron-wave propagation from a

quantum point contact to a quantum wire. Our numerical method solves the boundary problem of a lat-tice model, and determines wave functions at an arbitrary site. This method also includes a recursiveCareen s-function method. Our study found oscillations in the conductance, and magnetic suppression ofthose oscillations. For a simple model, we simulate the oscillations directly related to the channel num-ber in the quantum wire. To understand the magnetic suppression, we investigate the dependence of theelectron-wave propagation on the magnetic field using a realistic model. Numerical results show that arealistic rounded corner at the point-contact and a magnetic field could suppress the oscillations. Wealso discuss the transition from a classical skipping orbit with clear circular segments and focusing to aquantum edge state along a potential wall.

I. INTRODUCTION

Mesoscopic systems have interesting electrical trans-port characteristics due to their coherent ballistic elec-trons. For example, we can give the following: conduc-tance quantization (in a point contact), ' magnetic focus-ing, and Hall effect for a one-dimensional (1D)crosswire. ' These mesoscopic phenomena have beenmeasured in an artificial system with a variable confiningpotential. Many systems have been fabricated usingsubmicron lithography and high-mobility two-dimensional (2D) electron gas structures. In these sys-tems, the dimensions are significantly smaller than boththe mean-free path and the coherent length at low tem-peratures. There have also been many theoretical studiesof the mesoscopic phenomena of electron-wave interfer-ence and ballistic trajectories. Conductance quantizationhas already been explained clearly, ' and numerical cal-culations have shown the ideal skipping orbit for magnet-

ic focusing. ' Numerical analysis has pointed out thatthe shape of the device strongly afFects its transportcharacteristics. For example, the Hall effect in a 1Dcrosswire is defined by the shape of the crossing. ' '"Other examples have predicted the chaotic resistanceAuctuations due to scattering from a geometricfeature. ' '

In this paper, we present a numerical analysis of ourexperimental data for the mesoscopic system. We detailthe dependence of the transport characteristics on thesubject's shape. We measured the transport characteris-tics of a device with a point contact and a reflector. ' Fig-ure 1(a) shows our device and circuit when we measuredthe point-contact resistance. The device was fabricatedon a modulation-doped Al Ga& „As/GaAs heterostruc-ture epilayer. The mean-free-path length was 4.5 pm andthe Fermi wavelength was 44 nm at 0.34 K. This device'sdimensions are less than the mean-free-path length; theelectrons are expected to move in this device nearly bal-

E$, I

lfI I ' 51

I I II I I4 15

gy

~m~r~r~~r. .~r' '4 4 i (! er5544555e~

I,.I. I 55 5 44j I I"=555 5 44 5 I» 555555~III!55 I. . . . . , . I. l .5

)&

3 pm

Split-gate Reflector gate

P6int contact

~)~ «5

)

Electron w

FICi. 1. (a) A scanning elec-tron microscope micrograph ofour device with control andmeasurement circuits for point-contact resistance. (b) Diagramof a confinement structure witha point contact and a reflector.The shaded region indicates aquantum wire with 1D channels,where the 1D channel number iscontrolled by changing the volt-age V&.

Depletion region

0163-1829/95/52(11)/8244(12)/$06. 00 1995 The American Physical Society

52 NUMERICAL ANALYSIS OF BALLISTIC-ELECTRON. . . 8245

listically. The measurement temperature and appliedcurrent were low; therefore, the phase coherence length isexpected to be comparable to the device's dimensions. '

Figure 1(b) shows both gate structures and the depletionregions. The electron gas system confined by these de-pletion regions has a T-shaped configuration that consistsof a point contact and a quantum wire along the reAector.The point-contact opening was fixed, as it has a max-imum of two channels. That is, we fixed the voltage ap-plied to the split gate. The shaded region in Fig. 1(b) isthe quantum wire, and its width was controHed by chang-ing the voltage V~ applied to the reAector gate. Accord-ingly, our devices have only two geometrical parameters;the width of the point contact, and the separation be-tween the point contact and the reflector. These parame-ters could be evident from measuring the conductancequantization. This advantage allows us to carry out amore detailed analysis of the experimental results than inother device structures. ' ' We measured the point-contact resistance with changing Vz by using the circuitin Fig. 1(a). For monitoring the channel number in thewire, we also measured the four-terminal quantum wireconductance, where we coated two terminals at the leftside in Fig. 1(a). The point-contact resistance in Fig. 2(a)

18

is periodic where there is no magnetic field, and theperiodicity relates to the channel number in the wire, asevident from the quantum wire conductance in Fig. 2(b).The magnetic field completely suppressed the oscillationat 0.2 T. The oscillation was suppressed, but the conduc-tance quantization of the wire remained, even in a mag-netic field. '

To analyze these properties, we numerically investigat-ed electron-wave transport of a T-shaped structure in amagnetic field. A detailed analysis required a calculationmethod that could obtain both conductance and spatialinformation of a 2D electron system with complex poten-tial profile. Our numerical analysis used a method forsolving the boundary problem of a lattice model, and forobtaining wave functions at an arbitrary site. A methodhas already been developed to study mesoscopic phenom-ena for a line-shaped, point-contact injector and awedge-shaped, point-contact detector. ' We apply thismethod to the lattice model in our paper. This methodincludes the recursion using the Green's function. '

Details of the calculation method are given in Sec. II. InSec. III, we apply this method to our geometry with anideal abrupt configuration, and we discuss the clear con-ductance oscillation. Section IV gives numerical resultsfor a realistic structure, and the numerical results showthe magnetic suppression of the oscillations. Numericalcalculations also show an interesting magnetic property,a transition from a classical skipping orbit to a quantumedge state with changing magnetic fields. A summary isgiven in Sec. V.

15II. CONDUCTANCE CALCULATION

IN 2D LATTICE MODEL

100.2 T

We can investigate the transport phenomena in mesos-copic systems by using the Landauer-Biittiker formu-la. ' To obtain the conductance, we have studied a nu-merical method for calculating stably transmissioncoe%cients. ' We now explain how the method expandsto a 2D square lattice model by considering a quantumwire.

The quantum-wire lattice model consists of a scattering

15

CO

10 Ideal wire

mi)

Scattering regionIdeal wire

-2.5 -1.5 0

IhII

~ !~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~V ~

"'":,:'. "::,t,: .'::.» ":::,'".'!'':::::.:N::. . : ':::".!':::-:::: i:!!»,' " ''!".".:;,' '

!:::.::.»»,:::!i!':";:.'::,:,. '.. ::.!«««», VI»!»'« «V««h»»I»

ff». ::,'::.':!;:::::-:., «-',:::C!':::::-".I::,:,:i ."' '.,-::I:!:.,-',:,:,.I:!:. ::::::i::::::! CC'i»' ' «i «i«»V', »",'»««V«!i«i»e~»

. I '«'*V«' VV»V! I» Vh»«' »V«»!':: '',-:,;;:. :.I!I '!»:!".i:::...„.!

&:':;.";;',:.,'::.&'::,': ';i-,:.".;:,"!:."'::'.:-'i'" .","!!i:i' 'i' '::;..;1. :.:,'!':,',',::::!»!::,',,!!,",-',»,

!I':.!:':::::':.i!.::::.'::.. :.::h': '''':: ': '::.».»-,.R4,,'I."..—. il .44»i«I.!:»i~i

''«i'»Vh!"I'»! "'vv'»I»«'iS'i»". «ii'» «'Vi'«)'»' ««'4

. :'-:.': -ISI»+h!I »V «»'ik! I!!«I»!V VSV!V «»iV I»+f,

IIII II I

I~ ' ~ h

IIIIII

V» ~ ~IIIIII

IIIII

III

'»

~ «IIIII

VR (V)

FIG. 2. Experimental results at 0.31 K with changing V&. (a)Point-contact resistance with osci11ations. (b) Conductance ofthe quantum wire, where we floated two terminals at the leftside in Fig. 1(a). The conductance quantization means monitor-ing the 1D channel number in the wire. The oscillations arecompletely suppressed by a magnetic field. Nevertheless, theconductance quantization remains while in the magnetic field.

y(0)' yj 1 )' ...'y(j- 1 ), , y(j)"'. ..:,

'

y(iv) yjN+1).:"g ( )~ »» ~ J ~

~ ~ . '. . :,:, . !,,:::;»-,,:::, !.i:-:::: .'.;!!«I.''!I:., '::::,". ::-, i»I'»:."=: '::."":-.!.:» ':-I.:.:,:,''4I» II I ~ ~ II I ~ ~ II I ~ ~,'. .:::"."' '. :::.':!;;:.::.' ': -"!!"';:.;., '. .''I'':".'."'''.":, ::;-". :::...C ';:;:::,:':,',:! I II I ~ ~ I II I

» ~ J 1 ~ ~ h» ««!'V V»»!!«»' «»V'4' «»i» «P» «!IV! «!»» «»V!» '«'»»»«i»!»»I«»V JW4» J JI 1 » ! I I

II I ! -:. : .:,",. .!' "':.': . . -'.!,' ',".':-:.:.','!:::.:-:.,' . ::.;:::::.:0:: 'i::,.i;:.:,!i ';::::. .;»;: '.'::::-:: !::!.. . ~ I II I '. ','.:.:»:,I; .'I . :!It; .'-':. ' ' !,::.:",;::::.!".:- . I I II I ~ . - . '-,'.::& --::,:::-::V'II', ,

:'.::::::-::".!.::.':-"::,I;,'.::.;..!!'-':.':,:.:,!.:' ' ':.':-;. I&-.:,':, ;:: . '.::::,'.:''.:~ II I !'.': .!': -':.:!' ';. ".'ii:.::!-»""::.' !!::::-., :;::..!$::-:. I I

0 1

FIG. 3. Configuration of a lattice model for a quantum wire.Integer coordinates (I,m) mean lattice points, and the latticeconstant is taken as a unit of length. This model consists of ascattering region and two ideal wires of width M+ 1.

8246 USUKI, SAITO, TAKATSU, KIEHL, AND YOKOYAMA 52

region and two ideal wires with a width of M+1, asshown in Fig. 3. The coordinates (I,m) are the latticepoints. The lattice constant is taken as being a unit oflength. The Hamiltonian of this model is given by

g al (Hl EF)al+al Hl, l+ial+1I

(2.1)

where al is the column vector of annihilation operatorsal for a site state al= (ai„.. . , aiM). Here, Hl, Hl I

=Tand 8& I + &

are M XM matrices defined by

y(0)

y(1) 0 L( —)=T

y( I) y( I —1)

y( I + 1) I y( I)

R.(+ )T—1

R( ) 0 y(N+1)

The matrix Tp is given by

for 1 ~ l ~ X,

(2.8)

(2.9)

(2.10)

Vii+4t

0

V„+4t0

VI3+4t

Vav+ 4t

(2.2)

U(+ ) U( —)T =

0 U(+ )g(+ ) U( —)g( —)

with

and

U(+) = [u, (+), . . . , uM(+)],

A,(+)=diag(A, ,(+), . . . , &M(+)) .

(2.11)

(2.12)

(2.13)

and

Hl I,=diag[ t ex—p(2niB), . . . , —t exp(2niBM)],

(2.3)

(2.4)

u is a column vector describing the amplitude of themth channel for M sites along the m axis, and A, is aphase factor of a plane wave along the l axis for the mthchannel. The column vector and the phase factor aregiven by solving the eigenvalue equation below:

B=1/XFR, . (2.5)

In the following numerical calculation, we set A,F to 10,since this is long enough for a lattice constant. The Fer-mi energy E„andkF follow

In Eqs. (2.2) and (2.3), the nearest-neighboring integraland site energy are given by —t and V&, respectively.The magnetic field B is included in terms of a Peirls phasefactor. B can be specified by a Fermi wavelength A.F anda cyclotron radius R, :

0 (+)—H(),'H(), H0,'(EF H() ) & (+ )u (+ )

(+)=A~(+) ~ (+) (+) . (2 14)

In the following discussion, the column vector u is nor-malized as ~~u ~~=1. The matrix Tl for 1~I ~%is givenby

02'E =2t 1 —cos (2.6)

—Hl I'+,Hl I, HI I'+, (EF HI)—(2.15)

The wave function 4 in the system is defined by theabove Hamiltonian as H ~%') =0. In particular, 2M chan-nels are formed in the ideal wires since the site potentialVI in the region of the ideal wires is not dependent onposition /. Then, to study the scattering problem of thewave function, we consider the channel coefficient L(+)in the ideal wire on the left and R(+ ) in the ideal wire onthe right, where + ( —) refer to waves moving to theright (left). These coefficients are represented by acolumn vector with M elements, and satisfy the followingequation:

2 2

m, n m

The matrix t of the transmission wave is defined by

(2.16)

The whole wave function from one outer region toanother is simply solved by using Eqs. (2.8)—(2.10). Wecan also obtain the transmission probability (t„)fromthe incident channel m with velocity v to the outgoingchannel n with velocity v„.In terms of the transmissionprobability, the conductance 6 of the system is obtainedfrom a multichannel version of the Landauer-Buttikerformula below

R(+ ) L(+ )

R( —) L( —)

(2.7)0 =TP TN

ITp ~ (2.17)

We define f(I) as a column vector describing the wavefunction of the lth cell consisting of M sites. The(2M X 2M) matrix Tl satisfies the following:

Note that the above equation is extremely unstable ingeneral. We stabilize it by using the following iterationtechnique:

NUMERICAL ANALYSIS OF BALLISTIC-ELECTRON. . . 8247

C(l +1) C(1+1)1 2

C(l) C(1)1 2

Tl 0 1 Pl for 0(l(%+1(2.18)

with

and

TN+ 1

,0 [U(+ )A,(+ )]1 —U(+ )[U(+ )A(+ ) ]

(2.19)

1 0ll 12

(2.20)

An initial condition of the iteration is C', '=1, C2 '=0,and T&+1 is added to Tl. Matrix Pl is a linear operatorto satisfy the form. that is assigned to the matrix on theleft of Eq. (2.18). Then we treat this linear operator foreach connecting step in the computation. The detailedexpressions are given by

P11 P12T121C1(1) (2.21)

l2 [ l2)C2 +T)22]

where

(2.22)

111 112

T121 T122(2.23)

This iteration continues from 1'=0 to N+ I, and it finallygives t=Cl '. A similar iteration gives the matrix r ofthe reAection wave as

the well-known Green's-function method' ' (see Ap-pendix). The advantage over the Green's-functionmethod is that the transmission matrix, reQection matrix,and wave function at any point can be obtained simul-taneously by the linear operator Pl. Calculating the posi-tion probability density is important in understandingboth electron wave propagation and mesoscopic devicedesign. The numerical calculation of Pl spends the samesize as obtaining the Green's function in our case.

We would normally investigate the electron-wavepropagation in our experimental results using thisfeature. However, the lattice model of the simplequantum-wire structure has to be extended to a structuresuitable for our device. As shown in Fig. 4, we use a T-shaped structure consisting of a wire A of width L+1and a wire B of width M+1, so that the lattice size isdefined by two parameters, L, and M. Wire 3 and wire Bare related to our device's point contact and quantumwire along the reflector in Fig. 1(b), respectively. The T-shaped structure is a kind of three-terminal device. Toanalyze the transport of this structure, we should use thethree-terminal Landauer-Butiker formula in general.However, for the point-contact conductance by our cir-cuit in Fig. 1(a), the three-terminal formula can be re-duced to the two-terminal formula, because the chemicalpotential of a terminal in wire B is equal to that of anoth-er terminal in wire 8 (see Fig. 4). Then we use Eq. (2.16)where the outgoing channels are defined by channels to-ward two terminals in wire B. The iteration method canalso be applicable to the T-shaped structure by extendingall matrices and changing suitable gauge when /= 1.. Inthe following section, we show numerical results fromour method.

( D( I + 1)D()+ 1 }—

( D( l)D(1) )P and r —D(N +2)1 2 1 2 I 1 (2.24)

where the initial condition is 0',"'=0, D~z"'=1. In orderto obtain the electron density at a point, we define a rowvector and calculate as follows:

(y(,"(l,m), . . . , y(" (l, m))—= (0, . . . , O, S „.. . , 6

(2.25)Wire A

mls g(+) g(-)

I I I

---. )IIIIIIIIIIIIIIIIII II I I IIIIIIII I+IIIII - - - --

and

(1l', "(l,m ), . . . , 1l'J+"(l,m ))

=(gI~'(1, m ), . . . , g'z~~(l, m ))P~ for l ~j«N+1 .

(2.26)

Then the position probability density is obtained by

I I

'M.::II

~ ~

1~

4~ ~~ ~

-e---''I ~

~ ~L::;::oaf'l- eat'&--II- I

~ ~ v\ ~ I

0 11 p

Wire 8 'll III)IIIIAIIIIIIIIII)IIIIAIIIII----I I

n (l, m }=~

f' + '(l, m )~

(2.27)

where the parameter i refers to the left channel number,the injected channel.

The iterative technique is useful for obtaining stablecalculations, because it cancels divergence factors thatoccur due to the rounding errors of each step. When themethod is slightly modified, it becomes as applicable as

Wllllllllll)Ill llll IIllllll)llllklllll - ----I

I

I

J

g(+) g(-)

FIG. 4. Configuration of a T-shaped structure lattice modelfor our experimental device. This structure consists of a wire A

with width M+ 1 and a wire B with width L+ 1. Wire A is thepoint contact in Fig. 1 and wire B is the quantum wire in Fig. 1.


III. T-SHAPED STRUCTUREWITH ABRUPT CONFIGURATION

We measured the oscillations of conductance for apoint contact with a reflector as shown in Fig. 1.' To un-derstand a property of the oscillation, we initially try cal-culating the conductance of a T-shaped structure with ahard-well potential and an abrupt configuration initially.It seems that the complex multiscattering of electronwaves rarely occurs in this structure, compared to a real-istic structure. Therefore, we can easily derive a simplepicture of the oscillation in the simple T-shaped struc-ture.

As shown in Fig. 4, the T-shaped structure has twobasic parameters, L and M. Wire 3 of width M+ 1 is re-garded as the injector of a point contact in Fig. 1. Pa-rameter M is set M=20 with A,+=10. Wire B of widthL + 1 is the quantum wire made by the split gate and thereflector gate in Fig. 1. Changing parameter L refers tochanging the applied voltage of our reflector. Figure 5shows the relationship between the conductance, fromthe injector to the infinite wire, with changing L. Whenan electron wave is injected through the first channel inwire A, in Fig. 5(a), the conductance oscillation corre-sponds well with a period of forming channels below theFermi level in wire 8, as shown in Fig. 5(c). The angulardistribution of the injected wave through the second

1 I

Total~I0.8-

0.6—

0.4—

0.2—

0.08

0.04—7th

0.026th

00' —5th

4th

0.0023rd

0.001 —2nd

1st

500

45 55 60 65

LFIG. 6. Conductance contributing to each channel in wire B,

where the broken line shows the total conductance. The period-ic oscillation is mainly defined by matching the injected wavefunction against a channel just below the Fermi level.

0.8—

I

(a)I I I I I I

Injection through 1st channel

channel is wider than through the first channel, ' ' be-cause the kinetic energy for the confining (fiowing) direc-tion in the second channel is higher (lower) than the firstchannel. Accordingly, by injecting the wave through the

0 I I

O I I

I —(b)o 08—

0.6—

0.4—

0.216--

Injection through 2nd channel

'

'(J' I

L=50

ll

ll, 6 lA! g~

12—

8C4

30 70

I

0 I I I I I

10 50 90L

FICs. 5. Conductance from wire 2 with M=20 to wire B bychanging L, for an ideal abrupt structure. We set a Fermi wave-length A,+=10. (a) Injecting electron waves through the firstchannel and (b) through the second channel. (c) The channelnumber in wire B. The conductance clearly shows an oscillationfor a period of forming channels below the Fermi level in wireB.

(b)

ilsP~ 2-

!)))

0-0 1 2 3 4 5 6

I/A, FFIG. 7. Position probability density for the ideal abrupt

structure. (a) L =50; matching case. (b) L =53; not matchingcase.


second channel in wire 3, the conductance, of which theoscillation becomes smaller, is higher than through thefirst channel [see Figs. 5(a) and 5(b)]. We think that theoscillation is caused by a matching between the channelsin the infinite wire and a wave function consisting of in-jecting and rejecting waves. Figure 6 shows the conduc-tance contributing to each channel in wire B, related toFig. 5(a). As is evident from Fig. 6, the total conductanceand the oscillations are mainly defined by matching theinjected-wave function with a channel just below the Fer-mi level. ' The conductance oscillates for a period of halfa Fermi wavelength. The distribution of electron wavesin real space helps in understanding the ballistic trans-port of a mesoscopic system. Figure 7 shows the positionprobability density L =50 (matching case) and L =53(not matching case). This figure shows that maximumconductance occurs at matching channels in wire B, andthe mode formed by injecting and rejecting waves.

R A, =1We calculate the conductance in a magnetic fi ld 'th

, / z=l0. This corresponds to 0.2 T in our experi-ment. In Fig. 8(a), the oscillations become slightly small-er than in no magnetic field. When an electron wave isinjected through the second channel in wire 3, the oscil-lations become larger than in no magnetic field in Fig.8(b). Obviously, these results are different from the mag-

netic suppression in our experiment [Fig. 2(a)].e obtain a simple picture of the oscillation in the

matching electron waves. However, the simple T-shapedstructure with an abrupt configuration cannot explain allthe properties of our experimental results.

IV. PROPAGATION IN REALISTIC STRUCTURE

Previous numerical studies obtained the propagation ofelectron waves for only a simple abrupt structure, com-posed of wires with hard walls. ' The propagation in real-istic structures has not been investigated thoroughly, al-though we know it is sensitive to the shape. The follow-ing numerical results clearly show that the interpretationof an actual experimental ballistic transport needs calcu-lation that account for complex multiscattering. We ex-amined a T-shaped structure with a realistic potentialprofile.

A. Conductance oscillations with rounded shape

It has been reported that the confinement potential ofthe quantum wire caused by a split gate has a parabolics ape. Based on our previous report, ' we assume that

(a) 1F

O.g—

0.6—

0.4—

~ 0.2—o 0

(b)0.8—

0~ 0.6—(I,m )

0.4—

016--

I

Injection through 2nd channel

I I I I I

I I I I I

10

C8

CC

4

90300 I I I I I I

10 S0 70L

FIG. 8. Conductance from wire A with M=20 to wire B bychanging L for an ideal abrupt structure in a magnetic field withcyclotron radius R, /A, F = 10. (a) Injecting electron wavethrough the first channel and (b) through the second channel. (c)The channel number in wire B. The oscillations remain in themagnetic field, which is difFerent from our experimental results[see Fig. 2(a)].

0-2 2 4 6I/1p

FIG. 9. (a) Schematic picture of Eqs. (4.1)—(4.4). (b) Topview of a realistic structure potential with 8'=141, R =62

p

L =100, and M=80. The potential energy is taken as a unit ofFermi energy EF. Note that we did not draw higher potentiallines than V=2, since these lines would make the figure com-plex.


V =A. ~(m —R )(m+R —M —1)+1, (4.1)

when l, m are in region I in Fig. 9(a),

V =1—A,W —(2m —M —1) r —R (r —R),W+(2m —M —1) ~

(4.2)

wire A (for a point contact) and wire B (along a retlector)have a lateral potential profile composed of a parabola.We also assume that the connection between wire 2 andwire B has a rounded corner. As shown in Fig. 9(a), wedivide the T-shaped structure into several regions, andthe potential configuration has symmetry alongm =(M+1)/2. A detailed potential profile to the fourregions is given by the following expressions, for whichpotential energy is normalized by Fermi energy:

0.40.2

00.4

0.2—

0+o 03o

0c 0.200

0.2

00.01

7th

'...6th5th~1

i ~g

'~

1'~ r

ISt I

.. 2nd

4th3rd ---. .. ~ '. .---r

Total

12th

10th

~~m =~F' L+12

L+1 —l —R —kF

when l, m are in region II,

Vi =A,F (r —R XF)—when l, m are in region III, and

2

(4.3)

115 120 125 130

FIG. 11. Conductance contributing to each channel in wire Bfor a realistic structure. The periodic oscillation for a realisticstructure is not only defined by matching the injected wavefunction against a channel just below the Fermi level.

(4.4)

I' -(a)0.8—

0.6—

o.4—

0.2-„"'-(bi0.8—

00.6—

Injection through 1st channelI I I I I I

I I I I I I

0.4—

6 -(c

Injection through 2nd channelI I I I I

12

8Cg5 4

80 100 120 140 160

FICr. 10. Conductance from wire A to wire B by changing L.We set the Fermi wavelength A,F =10 and assumed a realisticstructure with W=141, A~=62, and M=SO [see Fig. 9(b)]. (a)Injecting electron wave through the first channel and (b)through the second channel. (c) The channel number in wire B.The conductance clearly shows oscillations for a period of form-ing channels below the Fermi level in wire B, when the parame-ter L = 100—140 (115—140) for case (a) [case (b)].

when l, m are in region IV, wherer =Q(l —I ) +(m —m ), and I =m = —(W —M—1)/2. In other regions, V& =0. The potential of apoint contact has two main parameters, R and O'. Infig. 9(a), R~ is the radius of the rounded corner, andR, 8' relate to both the metal gate structure and thep7width of the depletion region in the 2D electron gas. Inthis paper, we assume 8'=141, R =62, with A,F=10.These values are suitable for a device with about a 0.2-pmgate length, 0.4-pm gate opening, 40-nm Fermi wave-length, and two channels in the injector below the Fermilevel. ' These dimensions are reasonable for most mesos-copic experiments. Figure 9(b) shows a top view of apoint-contact potential with the above parameters. thepotential profile along the m axis for the injector is com-pletely parabolic. In the following, we use a specificvalue, R~/A, ~ =6.2, as a typical case of a second conduc-tance plateau for the injector. '

Figure 10(a) shows the conductance as a function of L.It is more complex than Fig. 5. When the openingaround a connection of quantum wires is sma11(L =80—100), the conductance oscillation is random.This might be similar to random transport of the so-called "chaotic cavity. "' ' ' When L ) 100, the oscilla-tions are clearly periodic. We then investigate the con-ductance contributing to each channel in wire B whenL =115—130. As shown in Fig. 11, a channel just belowthe Fermi level contributes to the total conductance. Theconductance also distributes other channels under Fermienergy more widely than in the case of Fig. 6. In Fig. 12,we show the position probability for the realistic struc-ture. These figures show the differences between L =116(matching case) in Fig. 12(a) and L =119 (not matching


20uV

18 L=119

16

14

12

10

') 0

IIII III 8

FIG. 12. Position probability density for theideal abrupt structure. (a) L =116; matchingcase. (b) L = 119;not matching case. Distribu-tions of the densities have complex patternsdue to multiscattering, but we clearly distin-guish between cases (a) and (b). The shaded re-gions indicate an electron density greater than0.05, since the contours would make the figurecomplex.

4 6 8 10

-(8)

0.8

0.6—

C 0.4—O I I' -(b)

0.8—

8 8 — J.I

16 -(c)'

Injection through 1st channel

h 2nd channel

case) in Fig. 12(b). They also show the complex scatter-ing phenomena. The realistic geometry leads todifhculties in the physical interpretation of the oscilla-tions. Note that the complex scattering cannot suppressthe periodic oscillation.

Contrary to the properties in no magnetic field, therealistic structure can show a simple picture for the mag-netic suppression. Figure 13 shows the conductance un-der a magnetic field with R, /A, ~=10. The magnetic fieldeliminates the conductance oscillation whenL =115—130. This result is quite different from that ofthe abrupt structure, but it agrees with our experimentalresults. The complete suppression is caused by therounded corner at the connection between two wires. Inorder to clearly show the effects of the rounded corner,we show the position probability density in Fig. 14. Fig-ure 14(a) shows rejections at the reIlector still dominateelectron-wave propagation in the ideal abrupt structurewith R, IA,F = 10. ReIIections of the realistic structure inFig. 14(b) reduce in comparison with the abrupt struc-ture, because the magnetic field guides the electron wavealong the rounded corner of the realistic structure. Thisguiding effect becomes clear in higher fields, and causesthe suppression of classical skipping orbits. Thedifference between the abrupt and realistic cases is alsoimportant to analyze other experiments. ' ' '"

12C

4

10— B. Suppression of classical skipping orbitand formation of quantum edge state

0 I

80 100 120 140 160

FIG. 13. Conductance from wire 2 to wire 8 by changing Lfor a realistic structure in a magnetic field with cyclotron radiusR, /A. F=10. (a) Injecting electron wave through the first chan-nel and (b) through the second channel. (c) The channel numberin wire B. The oscillations are completely suppressed by themagnetic field for L & 110. This can explain our experimentalresults [see Fig. 2(ai]. However, random oscillations remain forL (110.

Under relatively higher magnetic fields, the ballistictransport has several interesting properties. The classicalskipping orbit with clear circular segments and focusing,and the transport by the quantum edge states, whichmean channels in a quantum wire under magnetic fieldshere are both well-known phenomena. The magneticfocusing experiment showed the existence of a classicalskipping orbit, and interesting results of numerical calcu-

8, 9lations. ' There have been no numerical studies of thetransition form a classical skipping orbit to quantum edge


20

is-Ib) Up p

U

-'OOp14

14

10

iz- Il(( ( 'pi

g J Lr ~v

ri q qU

iI'U'o,

0 00

UU(tJUP

4

FICx. 14. Position probabilitydensity for applying magneticfield with R, /A, F = 10. (a)L =140; the actual structure. (b)L=73; the ideal abrupt struc-ture. These cases have sevenchannels under Fermi level forwire B.

0 8 10 120 I I I I I I I I

0 2 4 6

state within changing magnetic fields. Our results clearlyshow that the transition in the realistic case is differentfrom the abrupt case.

Figure 15 shows abrupt structure conductance for vari-ous channels while subjected to magnetic fields. Due todeflection by magnetic fields, the rising current in wire Bis larger than the sinking current, even in the weak-field

case with E., /A, F=20. The contribution of the channeljust under the Fermi level remains large under a weakmagnetic field with R, /A, z = 10,20. The conductance (seeFig. 8) maintains oscillations, because the wire s width issmaller than the cyclotron radius. Under stronger fields,

10

(a)0.6 I I I I I I I I I I

0.5

Rc/1F ——[n] [20]

0.4

V0.3

V

0.2

[10]

[20]0.1

0 I

1 2 3 4 5 6 7 8 9 10

Channel number

FICx. 15. Distributions of conductance at each channel in amagnetic field for an ideal abrupt structure. The broken linerefers to the sinking current (see inset), and a bold line is the nomagnetic field case. For a higher field R, /XF = 1,2, the distribu-tions have peak structures, which indicates the formation of aclassical skipping orbit.

0

Injection through1st channel

4 0

Injection through2nd channelI I I

2

FIG. 16. Position probability density for an ideal abruptstructure in a magnetic field with R, /A, F =2. (a) Injecting elec-tron wave through the first channel when L =50,M=20 and (b)through the second channel. These densities reveal skipping or-bits with half circles and periodic magnetic focusing. The shad-ed regions indicate an electron density greater than 0.2, sincethe contours would make the figure complex.


R, /A, F =1,2, the distributions of the conductance contri-butions clearly have peak structure. This implies the for-mation of a classical skipping orbit. It is represented bythe plane waves of several channels, in terms of the wavemechanics. In order to support this explanation, weshow the position probability density in Fig. 16. The nu-merical results for R, /A, F =2 show obvious half circles,concerned with the cyclotron radius and the periodicmagnetic focusing. In the case of the abrupt structure,the classical skipping orbit could be formed when aninfinite wire's width is larger than the cyclotron radius.The conductance oscillation does not occur under thiscondition. Orbit skipping has been well investigated foran abrupt structure. ' However, the propagation of elec-8, 9

tron waves through magnetic fields in a realistic structurewith a rounded corner is different from the propagationin an abrupt structure.

Figure 17 shows the conductance contribution at eachchannel in magnetic fields for the realistic structure. Inspite of deAection by magnetic fields, the rising current inthe infinite wire is smaller than the sinking current whenR, /kF =20. This is that multiscattering for both round-ed corners and refIector makes rebounding of electrons. "Under stronger fields, R, /A, ~=2,4, the distributions ofthe conductance contributions have a maximum at thefirst (second) channels in wire 8, when electron waves areinjected through the first (second) channel in wire A.

20

18—

R lk —4

injection through1st channel

Inj2n

12—

This channel selectivity means the formation of an edgestate. The position probability density in Fig. 18 clearlyshows a transition from a classical skipping orbit to aquantum edge state with changing magnetic fields. ForR, /A, F=4 in Figs. 18(a) and 18(b), the magnetic field al-most guides the injected electron wave along the wall.The propagation, however, leaves both an incompleteskipping orbit and obvious magnetic focusing, ( speciallyfor injection through the second channel in Fig. 18(b).For R, /AF =2 in Figs. 18(c) and 18(d), the position prob-ability densities show formation of magnetic edge stateswith smaller undulations. The magnetic field withR, /A, F=2 achieves a strong channel selectivity for therounded corner configuration. The propagation results ofthe realistic structures are therefore quite different fromthe abrupt structure.

10—

0.4I

—(a)

0.2—

] I

I I I I I I I I I

20]

[1O]

I I I I I I I I I

channel)

nnel)

I I I I I I I

20— (c)

18—

Rc/1F 2

Injection through&4 1st channel

12—

10

I

— — (d)

Rc/il.F——2

Injection through

0.4—

0.2—

0 . I- . I

1 2 3 4 5 6 7 8 9 10

Channel number

FIG. 17. Distributions of conductance at each channel in amagnetic field for a realistic structure. Broken lines refer to thesinking current (see the inset in Fig. 15), and the bold line is theno magnetic field case. For a higher field, R, /A, +=1,2,4, thedistributions show channel selectivity, which indicates the for-mation of a quantum edge state.

I I I I I I I

-2 0 2 4 6 -2 0 2 4 6

l/1F l/AF

FIG. 18. Position probability density for the ideal abruptstructure in a magnetic field at L =100. (a) Injecting electronwave through the first channel when R, /A, F =4; (b) through thesecond channel when R, /A, F=4; (c) through the first channelwhen R, /A, F=2; and (d) through the second channel whenR, /kF =2. These numerical results show the transition from aclassical skipping orbit to a quantum edge state with changingmagnetic fields.


V. SUMMARY

We have described calculations based on a 2D latticemodel for electron-wave propagation. Our method is ageneralized technique based on the recursive Careen's-function method and allows the determination of bothtransmission/reflection coefficients and the wave func-tions at an arbitrary position. It provides stable numeri-cal results for both hard and soft confinement potentialsand for both sharp and rounded geometries.

We used this method to examine ballistic electrontransport in a magnetic field. The structures examinedwere a quantum point contact and a rejecting boundarywith both realistic soft confinement potentials and round-ed corners. We compared these results to those calculat-ed for the simple case of a hard confinement potentialand sharp corners.

With no magnetic field and large L, the conductancefor the realistic model is clearly periodic. For small L thetransport exhibited random fluctuations similar to thechaotic cavity. This is related to the strong lateralconfinement between the point contact and the refiectorin this regime. This effect has not been observed experi-mentally, probably as a result of a smearing of the com-plex multiscattering by inelastic scattering.

The results showed in detail the transition from a clas-sical skipping orbit with magnetic focusing, to a quantumedge state with magnetic guiding. The trajectory of theelectron wave may be determined by the competition be-tween the magnetic effect and the collimation effect of apoint contact. It is difficult to predict the trajectory indetail without a numerical calculation. Our results forsoft confinement and rounded corners show both magnet-ic focusing and clear channel selectivity for R, /A, ~ near4. In the case of hard confinement and sharp corners, weonly found ideal skipping orbits with half circles. There-fore, there is a clear contrast between the results forrounded and sharp corners. This effect for the roundedcorners shows that the experiment (especially near 1 T)for magnetic focusing and edge state requires detailedanalysis.

The major results of our study relate to the magnetic-field dependence of the conductance oscillations. For thesoft confinement and rounded corner model, the oscilla-tions were found to be strongly suppressed at small mag-netic fields, consistent with experimental observations.Although both the simple and realistic models show asimilar oscillatory behavior with no magnetic field, thesimple model cannot simulate experimental magnetic

properties. Consequently, rounded boundaries are essen-tial for the interpretation of the magnetic field depen-dence observed in this structure.

ACKNOWLEDGMENT

The authors thank Professor T. Ando of the Universityof Tokyo for his support in the University Industry JointResearch "Nano electronics. "

APPENDIX

(oIG"'IO) =(E,—H, )-',

(j I

G"'Ij ) = IE —H, —H. .

and

&&(j—1IG" "Ij—1)H, j

(J IG'"lo) =(J IG"'lj)H,;, -g(j —1IG" "Io) .

Then, this iteration method is a generalized calculationtechnique including the recursive Green's-functionmethod and the method in Ref. 18.

The following shows that iteration based on Eqs.(2.28)—(2.32) is also applicable to the recursive Green's-function method. By using the expressions in Ref. 21, wechange HJ to HJ, and the matrix in Eq. (2.13) is appliedto a wider region as

0—

H~ J~+,HJ 1, HJ J'+, (E~ —H~ )

for 0~j +%+1 .

Furthermore, when an initial condition is given as

CP'=(F '(+)—F '( —))U(+) and C' )=0

the matrices C'&+",C2 +"obtained by iteration relate to

the recursive Green's function

cIJ+' =(j G J Io)HO, (F '(+)—F '( —))U(+)and

C21+"=(jlG'1' j)HJ,J+1

Here, Green's function 6 is defined by the followingwell-known relations: '

*Fax: +81-462-50-8844. Electronic address:usukitty@flab. fujitsu. co.jp

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H. van Houten, C. W. J. Beenakker, J. G. Williamson, M. E. I.

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a


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