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Topological properties of the edge states scattered by a simply connected point contact
V. Marigliano Ramaglia, F. Ventriglia, G. Zucchelli
To cite this version:
V. Marigliano Ramaglia, F. Ventriglia, G. Zucchelli. Topological properties of the edge states scattered by a simply connected point contact. Journal de Physique I, EDP Sciences, 1994, 4 (11), pp.1743-1754.
�10.1051/jp1:1994218�. �jpa-00247029�
/. Phi'i 1Fiafit.é, 4 Il 994) 1743-1754 NO,,EMBER 1994, PAGE 1743
Cla,,it'icjtion Pfi_i'i-ii,i Ah in a( i,;
Topological properties of the edge states scattered by a simply
connected point contact
V. Marigliano Ramaglia, F. Ventriglia and G. P. Zucchelli
Dipartimento dl Scienze Fisiche. Universita' dl Napoli. N-F-M- e G.N,S.M. <CNR), Mosira d'oltremare Pad. 19. 801~5 Napoli, lialy
iR<,<.en.<'d 21 Apiil /994, ut iepi(,d l~ -Ii(',,' l994)
Abstract. The ~inguJanties of the pha~e of an edge ;taie scattered by a barrier interrupting a w ire
ii a pei~pendicular magnetic field are di«ussed. The tran;mis,ion properiies Dl such a ~tructure are expres~ed in terms of creation and annihilation of vortex~antivoriex pairs that give n,e to o,cillation; in the tran~mi,Sion coefficient. Vortice, are trapped under the borner causing o,cillations in the tran,mi,,ion that are related to ihe re;i~tance o,cillation; mea,ured in quantum point contact;, who,e penod depend, linearly on the m;ignetic field. We propo,e a iheory of these
oscillation, which differs from other explanation~ ba,ed on the Aharonov-Bohm ei'fect. An e~periment is propo~ed in which o,cillations in ihe tran,mis;ion are monitored by the barrier
voltage gate.
l. Introduction.
Recently great intere~t lias been attracted by the Aharonov-Bohm conductance oscillations in mesoscopic metals induced by an applied magnetic iield. A large variety ai such effects has been ~tudied in terms of ~ingle charge tunneling in nano;tructure; (1, ~].
In a previous work [3] we developed a model ior a quantuil~ point contact on which a
perpendicular magnetic field acts. We consider a two-dimen~ional non interacting electron gas, confined in a narrow strip interrupted by a barrier. The chosen confining lateral potential Vi ii
=
niwj( t~ i~ particularly suitable to describe the edge states due to a magnetic field directed along the =-axis. The barrier U(j') arise~ along _i'-axi~. There we introduced an approximation for the Lippman-Schwinger e~uation de;cribing the ~cattering of the edge ~tate~
by the barrier. An exact ;olution was exhibited of the approximated integral e~uation in the
case of a ~quare barrier. The il~ain feature of the calculated tran~mission coefficient was the appearance ai resonant tunneling in the quaniuil~ region, thon is when the energy is below the top of the barrier.
Here we show that the phase of the ~cattering states has ~ingularities inside and out~ide of the barrier, which are vortice~ and antivortices respectively. The presence of a retlected edge state gives rise ta antivortice~ caused by the quantum interference between the incident and the
retlected wave. A regime of ihe barrier traversal is characterized by vortices trapped under the barrier.
1744 jOURNAL DE PHYSIQUE I N° II
In section ~ we di~cuss the tran~mi~~ion properties of the barrier in terms of such vortices
and antivortices. We show the trajectories spanned by the centers of the vortices and
antivortices in the (.i, j') plane as the energy E of the edge states or the strength of the magnetic field changes. The vortices and antivortices are created (annihilated) in pairs at the barrier edge
when the transmission is at a minimum. In the resonances there are only vortices under the barrier whose total number increa,es with the magnetic field while decreases with the energy.
The antivortices move from infinity io the middle of the ;trip going from a resonance to an antiresonance.
In the section 3 the calculated transmission oscillations are compared with the oscillations of
the resistance of a simply connected point contact calculated in [5. 6], whose period is
proportional to the magnetic field [4].
The transmission oscillations are strictly related to the quantum traversai of the barrer. This feature may be put in evidence varying the height of the barrer by a suitable gate voltage. A~
the voltage increases, the transmission o~cillates when the kinetic energy is well below the top oi the b3rrier m a full quantum tunneling regime as shown in figure 9. Because the voltage gate affects only the charge contained in the barrier, holding all other parameterq fixed, we believe that there is no simple definite relation between the magnetic flux encloqed in some area under the barrier and the predicted oscillations.
2. Vortices and antivortices.
Let us briefly resume the results of our previous paper [3]. The energy spectrum of the edge
states of the parabolic channel i~ ~tructured in subbands given by
E~~(t)
= (n +( fi+ ~(kj.
(ij
Here is n~
= w~/w~, w~
= eB/nie being the electronic cyclotron frequency and
wjj the characteristic frequency of the confining harmonic potential and
k~
~
+ QI
is the kinetic jor each subband. The energy is measured in units of hwj~ and the lengths
in units oi, h/2 mwo. When the energy of the incoming edge state belon g; to the first subband
in
= OI, but lies below the bottom oi the next one (n
= ), the wave function which is solution of Lippman-Schwinger equation take~ the forai
~(.t, j'i
=
fi iW+ Ii') fin(£Y~,t.
+ yÉ) + q~ ~>)iij~(a~,t ykii
,/2"
lb~~')-A(Éi~b~~')
W~ Li ~
~~ ~
12)
where #~ (j, are the projections on trie incoming edge state iii
=
°
au (a ~,t
+ yÉ e'~' and
, 2 ~
the reflected edge state iii = Î u~(
a ,; yL e~ '~' whose overlap is A (k
,'~ =
~
exp(- y~k~ with
a =
(fi12 )"~,
y = , 2 n/(1 + QI)~'~
N° Il TOPOLOGICAL PROPERTIES OF THE EDGE STATES 1745
The exact solution for a square barrier of height Ujj and length L, placed between
y =
0 and y
= L, is
#~ = [(l +ia)(1-iu-ii)e~~'~~'- il -ia)(1+iu-ii)e~~~'~~'l
u~
#_ ~ [(l iule~" ~~'- (l + ia)e~~" ~~~] (31
A where
il + IJÎ) Ujj
~
=
2 ju sinh lKLl ici cosh (KL Il 1'
~
~ ~2
and
K
=
iÉu with a
= ,/ iii l~ A~ u~. (4)
Here iK behaves a~ an effective wavevector for the wave function within the barrier. When
K is imaginary there is propagation in~ide the harrier.
In reference [3] we obtained the following analytical expres~ion for the transmission
coefficient :
T il +
~ ~ ~j ~° ~ ~~(/)) j (K imaginary
~
A ' + QÎI Uo L sinh KL j
~ ~
~ 2 1 KL ~~~
The unexpected feature of T is the re~onant tunneling through the barrier at energie; under the top of the barrier. Figure ~hows Tas a function of il~ (da;hed fine), at a fixed value of kinetic
energy ~ lower than the height of the barrier. In figure the ratio N/Njj is also shown jfull fine) : N is the number of electrons contained in the barrier, given by
j+ ~ j~ ~ ~
iiA~
j sin 2kaL
~ ~"~~
~
~'
~
~ ~~'~' ~ ~' ~~ ~ ~
u 2 kaL
and Njj is the number of electrons contained in a segment of length L in the channel without the barrier, when a single edge ~tate propagates along it. Starting from il~ greater than a critical value n~, the transmission coefficient oscillates between the value of at the re;onance~ and
some minima at the antiresonances. At the resonances we have a very strong localization
within the barrier. The oscillations of N follow exactly those of T. lncreasing the field,
K from real becomes imaginary and one has propagation through the barrer for energies at
which it i~ classically prohibited. The transmission stops when the field ~o high that the bottom
of the fir~t subband fi
goes over E the edge state cannot propagate along the
channel.
Let us begin to consider the singularities of the phase 1J1,t _i') of the wave-function outside of the barrier for _1'<0. For n~ m n~~ the square modulus of the wave function is
(4i(-t, Y)(~ = ~~'~~F(t,-Î j,) F (,i, )
=
(cos~ + L.Î sin~
ii~( + c( iii~ sin~
? <._ iii iii sin (cos À sin 2 Év + <._ sin À cos ~ É-1>) 16)
1746 jOURNAL DE PHYSIQUE I N° II
5
U~=Z L=5 E=0.8
, ,
1.29 1.48 ( 1.88
(ii~ (
~i 11
~> ii
~i >i
~i ii
~i ii .
~i ii
o ~i ii
z i ii
à (Î Î(
-4n
(1ii
~i ~
i U~=Z.L=5 ~
~~~ ~'~o~~ ~~ ~-Zfi
i .
l '
, ,
i
,' ,
, , , , ,
'~_, '~-'
'
0 l
Q~
Fig. l. Fig. 2.
Fig. l. The number N of electron; contained in ihe b3rner
in unit~ ot Njj (tull fine wiih the scale on ihe
lefi sidej and the iran;mi~sion coefficient T (dashed fine with ihe scale on the nght ~idei a, function, of
i2~, ihat is at increa,ing tield. The numbers on the lefi oi ihe tmn~mi,;ion peak~ give the flux of the
externat magneiic field in unit, hile enclo~ed in the area S~ ? L/w~i (2 E/m )"~ that
i~ shown in figure 8b.
Fig. 2, The stnpe; in which the vortices and the antivoriice~ are contained. The centers of vortice~
(antivortice~) are indicated with the ~ign + ). The fine integral~ around the circles have the values
~hown.
o
"~,,0877 ',,1',1.lZ8
~Z x10 ~"--~
1"~, 0
'~m- ~)jç,___
33 10~~ 7 lÔ~ 3 (0~~
0 88
~ O.93
085 0873 O.984 0887
0998
~~~~
I.OOO
anUvorUces irajeciomes varying Q~ ai E=0 8
Uo"Z L=5 139
O.OZ 0.04 0.06 0.06 0
Fig. 3. The.t-y trajectories of the antivortices centers in front of the barrer. The trajectories of the
antivortex nearest to the lower strie of the borner has been magnified by a factor 10 in j-direction- The
numbers put aside the points give the corresponding value of D~.
N° Il TOPOLOGICAL PROPERTIES OF THE EDGE STATES 1747
with
À
=
tuL 1
=
~~~
<.~ =
~~ '
a a
The components of the gradient of1J are
v~ = ô~1J =
2 a ykc_ iii iii sin (cos À coi ? kV <.~ sin À ~in ? ky )IF v, = à,1J
=
k( (cos~ + c( ~in~ iii ~ cÎ iii ~ sin~ )IF (7)
The modulus of the wave-function vani~hes in an array of point j.i~~,, j,~~j In ) equally spaced along a line parallel to y-axis. We get these coordinates by solving the following two
equations, obtained putting the real and the imaginary part of ~ equal to zero
fli~
= iii (.1~,~, cas À cas Éy~~, (c~ iii (.i~~j + c_ iii (,t.,,~jII sin sin kv~~i = 0 Cl~
= u( 1.;~~, cas sin
k_1>~~j + (c~ ii( (,;~~j) c iii j.;~,~~II sin cas Éi'~,~, =
o j8)
For.t~~j we have
t~~j =
In itan~ j (9)
4 y ut and
j 11( (.;~,~,) co~
i'u~iIii
= arctan iiar loi
k (c.+ iii (.i,,uji + c.- ii (fou,i) ~in
At the resonances (À
=
Îar, with Î integer), this fine of nodes goes to infinity, .;~~i
= + oJ. In
ihe minima of the transmission, at the antiresonances, is À
= (2 + ar/2 and cos À
=
o and thew nodes are located at
c_ n
~
""~~ ~ y ak
~~
c.~
~°~~~'~
k
At the nodes the components of the gradient of the phase diverge in such a way that the line integral along an anticlockwise closed path C, winding once around one of the points (-"oui, .i'~ui iS
li.dl=-2ar. (12)
The property implied by the equation (1~) can be expressed by saying that we have antivortices centered at the points, (~;~,~j, ;~~~) whose topo/ngica/ chaijqe is We note that equation il ~)
is exactly verified only when 1' is fully contained in the semi-plane _i'mû, that is outside the
barrier. Otherwiw the approximations on the wave function derivatives at the barrier edge
introduce small discontinuities that yield an integral slightly different from ~ar.
These antivortices sustain the coupling between the incoming and the outgoing edge state. In tact a magnetic field directed along the = direction imparts to a classical free particle of
negative charge an anticlockwise rotation. Its quantum counterpart is the diamagnetism of the unconstrained Landau level~. The confining potential V (,1) opens the Landau orbits into edge
states whose diamagneti~m implies the localization of the incoming edge states on the left hand
side of the stop (,t. mû ), while the outgoing edge states Jean on the nght hand side
1748 jOURNAL DE PHYSIQUE I N° il
(,; m 0 of it. The barrier reflects the incoming edge state~, by forcing them to go bock with a clockwise rotation, that is in the opposite direction of the diamagnetic rotation. Therefore the
retlection is always accompanied by the antivortices. At the maxima of the reflection
(antiresonances) the line of the antivortices stays just in the middle of the stop. On the cc)ntrary, ai the resonances, when the retlection coefficient is zero, the antivortices line is
pu~hed away to infinity along the.t-axis.
On the other hand, within the barrier, ~ingularities in the phase of the wave function appear
only when À is real, that is when the field induces propagation under the barrier at
n~ m il~,. The function F (,t, v that gives the modulus of the wave function for 0 < 1, < L is F j,1,
_~,j = iii~ cos~(À jj,/L + (c_ iii c~ iii )~sin~ (À (i,/L ). (13) The component~ of the gradient of1J are now
v, = ô~1J =
? aykc_ fi( iii co, (À (i'/L 1jjsin jÀ (y/L 1))IF
u, = à,1J = kart( (<._ iii c~ iii )IF. (14j
The coordinates of the nodes of the modulus of the wave function satisfy the equation~
cos (À (>IL ))
=
0
t._ uj (.i) t.
~
fi( (.t
=
0 and are
c_
t~~ =
~ ~~~ In ~
jj (2 à 1)1 ~
~~~~
."ii'1~~l ~
~ ~
where < s < .V and the total number.N' of nodes within the barrer i~ just equal to the integer part of Àlar + 1/2. We note that.t.~~ is equal to.t~~~ at the antiresonance;, but equation il 5) is
valid for any value of while equation (11) holds only at the antiresonance~. Again the
coniponents of the gradient both diverge at the nodes and now we have
1>.dJ=2ar, (16)
1,
for any anticlokwise circuit C" which is contained in the barrer and winds once around one of the point~ (,t.~~, _i,~~j. The~e points are therefore centers of vortices with topological charge
+ 1. The positive sign of the topological charge means a diamagnetic behaviour within the barrier.
Obviously only one edge state propagates towards j<= +oJ beyond the barrier for
V m L, and its pha~e doesn't have singularities.
Let us briefly discuss the magnetic structure of the vortices and antivortices. The ~emiplane
i, mû and trie barrier region ~l mi'< L con be divided in stripes extending along the,i-axis and perpendicular to _v-axis- each one containing a single antivortex or a single vortex
respectively as figure ? shows. Equations (12) and (16) mean that if we tal~e the points
t,~~, _i<,~ or (.t~,~j~ y~~~j as the origin of a polar coordinate systems (i iJ [ the wave function in
these polar coordinates has the form
~ r, iJ f (1, 0 e~ ~ 7)
N° II TOPOLOGICAL PROPERTIES OF THE EDGE STATES 1749
inside the strip that contains the origin. In other words there the phaw of the wave function ii
equal or opposite to the geometrical anomaly ~l. The function J Ii, 0 is real and single valued.
The ~ign of the exponent is positive for the vortice~ jcentered in.t,~, j',~)) and negative for the antivortices (centered in t~,~j, j<~~j)j. Although the phase of the wave function has singularities
in these points, the magnetic flux a~sociated to each vortex is not equal to one quantum flux
our vortices are different from those of the order parameter in ;uperconductivity ilOI. in fact
we note that within each stop we have
lim J (,t 1) = 0.
1- ~ DD
The angular momentum Î=
=
ifi ô~ is not con~erved, but its mean value in the strip enclosing
the vortex is
2« Rif') 2w Rif,(
do dr içi~ f- 4~
=
fi = do di it~ii., Hi
+ o
Îi Îi Îi
+ ~ ~
do ô~~(R~(0)) t~(R(0), 0)) = = fiM, (18)
4
i~
because J is single valued. The imaginary term is negligible because f ii qmall on the
boundary R(0) of the strip the mean angular momentum in the strip i~ approximatively proportional to the integral M of the square modulu~ of the wave function over the ~trip.
Therefore each vortex (antivortex) carries a mean angular momentum of Mfij- Mh where
M is the number of electrons in the ~trip in which the vortex is contained. A magnetic moment of oppo~ite ~ign corresponds to thi~ angular momentum because the electron haï a negative charge. Therefore the vorticeq are diamagnetic, that iq their induced field is opposite to the
external field, while the antivortices are paramagnetic in the sense that the lines of force of total field tends to concentrate at their centers.
Let u~ begin to show what happen; when the magnetic field incre3w~ at con;tant energy E < Ujj. As long as n~ < n~~, the parameter a becomes purely imaginary and all the formula,
have to be accordingly changed. In any ca;e, turning on the magnetic field the fine of the antivortices outside of the barrier move~ from t
= + oJ towards the center of the confining
channel. The wave function decay; inside the barrier as far as À is imaginary and the incident
edge state is almoqt totally retlected.
At higher fields vanishes and becomes real. At À
=
ar/2 the first vortex-antivortex pair is created at
_,, =
0. From now on vortice~ appear without changing the total topological charge
the appearance of a vortex under the barrier is alway~ accompanied with that of an antivortex outside of it A new vortex-antivortex pair stems out from the ~ame point (tj~ = t.~~~ on the
boundary
_v 0 of the barrier just when À
= (~ Î + ar/? in the minima of the transmission (antiresonances). As the field increases the vortex enters the barrier moving around
1= 0 while the antivortices- the new one together with the others- run rapidly away
di~appearing at.i = +oJ. When À
= ar, the vortex reache; the middle of the barrer
j< =
L/2 and the first resonance (T
= ) appear~. A; the field grow~ the vortex moves toward the other side of the barrier
_,, =
L while all the antivortice; return from the infinity to the center
of the channel. The transmission T diminishes to its first minimum at
=
~ar/~ when the second vortex-antivortex pair is created at the lower edge of the barrier.
In the wcond resonance (À =~ar) the two vortices are symmetrically placed ai
_v = L/4 and
_1, =
3 L/4 inside the barrier at the same titre the antivortices are repelled away
again disappearing at i
= + cc. Dut of the barrier there i~ a single edge state, which is completely diamagnetic and fully iran~mitted. Other re~onances may be generated according
JOUR~AL DE PHl~l0l E1 -T 4 N' il ~OVEMBER iQv4 h
