Physica B 212 (1995) 299-304
Coupled edge-currents in a mesoscopic quantum Hall system Vipin Srivastava School of Physics, University of Hyderabad, Hyderabad- 500134, India
A proposal of the author that a sufficiently narrow two-dimensional quantum Hall system of mesoscopic length should behave like a Josephson tunnel junction due to weak coupling of the edge-currents has been developed further here to study how the 'locked-in' phases of the two edge-currents slip with respect to each other in space as well as in time under the combined influence of the magnetic and the Hall fields. The spatio-temporal variation of the phase-difference, it is found, can be controlled by adjusting the system current. At a suitable velocity some of the current-carrying electrons, which also move back and forth between the edge-currents under the influence of the phase-slippage, form closed loops. As the electrons go around in a loop once the phase difference between the edge-currents changes by 2n thus showing that the state of the system is a single-valued function of the phase-difference and that each loop encloses a flux quantum, hc/e. In this way our semiconducting mesoscopic quantum Hall system mimics a Josephson tunnel junction mainly due to the long-range phase coherence. The quantum interference effects are discussed as they show up in some experimental and numerical results.
The understanding that phase coherence of the wavefunction can be mentioned over mesoscopic length scales in resistive systems if inelastic scattering can be minimised has proved vital in explaining many remarkable results. This knowledge has inspired people to transmute certain effects and aspects, hitherto thought peculiar to superconductors only, into mesoscopic resistive systems. Josephson effect is one such example. Imry and others  made suggestions about possible connection of the AC Josephson effect with the quantum Hall effect under special circumstances. The author 12] for the first time pointed out that the Josephson effect was naturally imbedded in a narrow quantum Hall system of mesoscopic length. Long-range phase-coherence in the edge-currents was argued to be central to the setting up of an AC between the edge-currents when they were so close in a narrow Hall sample that they could get weakly coupled. The weak interaction between the edge-
currents, though detrimental to the neat quantisation of the Hall effect, gives rebirth to the Josephson effect in the new setting by appropriately locking up the phases in the two edge-currents. The phase difference between the two sides undergoes relative slippage due to the inevitable presence of the Hall voltage in the quantum Hall effect (QHE) set-up and gives rise to the AC Josephson effect. We dwell on this idea further to investigate the details of the variation of the phase difference in time as well as in space. This enables us to work out the condition under which the AC should be setup and helps us in understanding the possible quantum interference effects arising due to the mixing of the two edge-currents moving in opposite directions. We will recall Laughlin's  gauge invariance arguments given in connection with the quantized Hall effect to emphasise the significance of long-range phase coherence in the wavefunctions spreading along the edge-currents, We will then use this information to assign welldefined phase~ to the two edge-currents. This will enable
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V. Srivastava/ Physica B 212 (1995) 299-304
us to use Anderson's  unifying concept of phase slippage when two regions of well-defined phases are weakly coupled. The uncertainty relation between the conjugate variables, number of charge carriers and phase, and the Hamilton equations connecting the variations of these two help us in working out a phase-driven current expected to be flowing between the coupled edge-currents and also how the phase difference between the edgecurrents moves in space and time due to Hall voltage, system current and the magnetic field.
path. As an electron enters the system with a certain phase, its phase is altered on each elastic collision event but in a deterministic manner; the phase memory is maintained along all the possible paths it can take between x = 0 and x = L, and the current can be taken as a phase-coherent superposition of wavefunctions in all the paths. Thus we can represent an electron in an edge current by a wavefunction of the type C(r)exp(ict) where C(r) is proportional to charge density per Landau sub-band along each edge.
2. Gauge invariance and phase coherence 3. The number-phase uncertainty Since the long-range phase coherence is central to the effect being discussed here, we first recall how it comes about through gauge invariance in a two-dimensional electron gas (2DEG) system under magnetic field. The 1D free electron wavefunction ~k (say that of the edge-current) is simply multiplied by exp(ieAx/hc) in presence of a magnetic field A being vector potential of the field applied normally to the 2DEG in which the edge-currents are flowing in the x-direction. It is conceptually easier to consider Laughlin's geometry  which is equivalent to putting periodic boundary condition on the 2DEG in the x-direction. In the ring geometry the ~k will acquire an extra phase for x = L, the length of the 2DEG (i.e., when the circumference of the ring is traversed once), and break the gauge invariance of the canonical momentum of the current carriers or, in other words, break the invariance of ~k under phase change (or the single valuedness of ~k) unless hc A = n--.
This condition demands that the 2DEG ring should enclose an integral number of flux quanta the eigenvalue spectrum must remain unchanged as flux through the ring changes by hc/e. In this way the gauge invariance makes the edge-currents persistent currents in the Laughlin geometry by attributing long-range phase coherence to them - more explicitly, the phase of ff may change by elastic processes between x = 0 and x = L but it must change in multiples of 2n. Now if we come back to the rectangular geometry, which amounts to unfolding the ring and connecting the two ends in the x-direction to two electron reservoirs, we find that an electron which leaves the system at x = L does not reenter at x = 0, so there is no question of demanding the single-valuedness of the ~b. However, the long-range phase coherence due to the absence of inelastic scattering must ensure a well-defined phase c¢ in the wavefunction in case L is less than the inelastic mean free
Following Anderson  we learn that the superflow in the edge-current is better understood in terms of the following Hamilton equation for the conjugate variables N, the number of electrons, and the phase ~,
h ( d N / d t ) = OE/Oct.
Gauge invarianee tells us that the energy of an isolated portion of the edge-current should be independent of ct. So, if the phase of this portion is not coupled to that of its neighbours, then dN/dt = 0. Consider a pair of neighbouring portions, AA1 and AA2, in the two edge currents and let the mean phases ~1 and ~2 in these elements be coupled by some energy which is minimum when ~1 - ~t2-~0 = 0. We will see that this coupling energy is the means by which the link between the two portions establishes long-range order in ct in that if no current source is connected to the combine then ~o adjusts itself so as to minimise the coupling energy. Thus the states of the system for which tp differs from the value which minimises the coupling energy are the current-carrying states. The flow across the boundary between AA1 and AA2 is given from Eq. (2) as J =
1 0E(~o) h 0q~
We, thus, learn that coupling between the two edgecurrents can make the electrons go back and forth between them leading to the possibility of coherence between states in which the total number of electrons is differently partitioned between the two sides - j u s t as the phase coherence within each side means that the number of electrons is not fixed locally (ANA~t ~ 1).
4. The phase-driven current Following the simple analysis of gef. , if we take the two edge-currents as weakly linked through a coupling
V. Srivastava / Physica B 212 (1995) 299-304
constant K then we find that the energy on each side, which was + e V n / 2 (Vn being the Hall voltage) in absence of coupling, reduces by E~ = 2K (nl nz)l/Z cos ~o,
where nl and n2 are the number of electrons per Landau sub-band per unit area which carry the edge-currents on the two sides. From the Hamilton equation (3) we obtain, using Eq. (4), the current density flowing between the two edge-currents, J =
2 K ( n l n2) 1/e
= d~ sin ~0.
field normal to the 2DEG. In the following section we work this out to have a comprehensive picture of the spatio-temporal variation of tp. This will enable us to study the quantum interference.
sin ~o (5)
5. Spatial variation of phase difference The ~0 entering the expression (6) for phase-driven current should include spatial dependence arising due to movement of the charge particles and the presence of the magnetic field. Both these factors enter ~ through the canonical momentum p = mr + eA/c. For the @ = [email protected]
', V~b = @(iVy), and since np is the expectation value of - ihV, we learn that the canonical momentum and the phase gradient are related as
Now we recall the other Hamilton equation between the conjugate variables n and ~o:
p = hV~ = mrs + e A / c ,
d 0E h~-~pot= - ~ n '
where vs is the velocity of the current carrying electrons. Then the phase difference between points a and b on the same edge-current is
where OE/On is the energy required to transfer an electron from one edge to the other - this is eVn in our case if for simplicity we take n~ = n2. Consequently, we learn that the phase difference ~0 varies with time as (o = e V n / h .
Thus the phase difference between the phases cq and ~z (of the edge-currents) locked-in by K, undergoes a continual slippage due to the presence of the Hall voltage VH in accordance with the Josephson equation (7). This phase-slippage drives the alternating current J between the edge-currents in accordance with another Josephson equation, namely Eq. (5). Note that while in the use of superconducting tunnel junctions a voltage cannot be maintained across the junction without accelerating the supercurrent, in the present set-up the Hall voltage is an integral part of the geometry so the phase-driven current (5) is always accelerated. The coupling constant K should be proportional to the probability of leakage from one edge to the other which we earlier estimated following Streda et al. , as Pxx/(Pxx + Pxy), where Px~ and Pxy, respectively, represent magnetoresistivity and Hall resistivity. The leakage probability turns out to be typically 0.05 which can lead to a transverse current of the order of 0.5 nA when the system current is about 1 nA. Now it is important to realise that besides its variation with time the ~0between the two edge-currents also varies in space due to (a) movement of electrons in the edgecurrents, and (b) the presence of the external magnetic
We choose a gauge in which the vector potential has no y- and z-components: A = lAx(y), 0, 0]; vs has only x-component, so following Ref.  we get the phase difference between two points lying in the edge-currents 1 and 2 with the same x-coordinate, (Pl2(X) = (p02 - - ( 2 m / h ) G x
where ~o°2 is the value of the phase difference at x = 0, B is the strength of the magnetic field, 6 is the separation between the points in edges 1 and 2 for which ~o12 is calculated and q~0 represents a fluxon. The second term on right-hand side of Eq. (10) makes a contribution smaller by more than two orders of magnitude as compared to that made by the third term, so we will neglect it . Combining Eq. (10) with Eq. (7) we get the spatiotemporal variation of ~o12(x, t) between two points lying perpendicular to the x-axis at the two edges of the &wide strip, which fall in the edge-currents 1 and 2, respectively, ~012(X, t) = ~012(0 , 0) -J- (2n/q~o) (Bx6 + Vat).
Note that if q~12(x, t) is plugged into Eq. (5) we will find that at a given instant J alternates periodically at definite points in the x-direction in accordance with the variation of q~x2in this direction; also at a fixed x the J alternates in time. At a particular instant, say t = 0, q~2 varies with X as (~O12(X, 0) = (P12(0, 0) -1- (2rt/d?o)Bfx,
I~ Srivastava / Physica B 212 (1995) 299-304
so the direction of the flow of the transverse current will flip at an interval Ax, given by
.... I Y
~o12(x, O) -¢p~z(0, 0) =nzt =(2rc/q)o)B6Axn,
n = 1, 3. . . . (13)
(note that the directions of J at ~p~2(x, 0 ) = 0 and ~0~2(x + Ax, 0) = 0 + 7t are opposite of each other with the IJI being the same at these two locations). Similarly, for a given x, ~o~2 varies with t as
~o12(x, t) = ~o12(x, O) + (2~t/Cko)Vnt,
and consequently the direction of flow of J alternates at interval At as
(])12(x, 0) =/'/7~ =
Now, consider an electron which is at x = 0, one end of the system, at t = 0 (Fig. 1) and suppose, for convenience, that q)12(0, 0) = ~002 =
(the main result is independent of this assumption). Since in this situation the maximum phase-driven current, Jc will be flowing from side 1 to side 2 [2, 6], the electron will be moving from side 2 to side 1 with maximum probability . This flow will continue until t = ~bo/(4VH) (the position coordinate remaining unchanged as x = 0) when J will be zero and the electron will lose all its y-momentum. At this stage the electron will begin to move along with the edge-current on side 1. In the time interval (0, ~bo/4VH) the distance travelled in the y-direction is
Fig. I. Above narrow 2DEG in a 4-probe geometry with edge-currents separated by a small distance w; phase-driven current flows over a distance 6. Below (a), (b), (c) spatial variation of transverse current J at t = 0, ?po/4Vn, and ~bo/2VH; ~,~ indicates position of an identified electron at these instances; J varies sinusoidaly as depicted by arrows of different lengths, the strength being proportional to the transition probability in the particular direction shown- it is maximum for qh2 = n/2, 3~/2, etc., the shaded rectangle of size 6 x (2Ax) is the loop which encloses one ~0o.
would cross over to side 2 with maximum probability (the transvere current at x = 2Ax being - Jc). In this case, we shall have ~o12(2Ax, ~bo/2Vn) = rt/2 + (2n/~bo) (Bf2Ax +Vn(~bo/2 Vn)) = 7 n / 2 ,
6 = vrq)o/(4Vu ),
where the vy is yet u n k n o w n - we will make a plausible estimate later on. At t = ~bo/2 VH the electron would have moved a distance Ax in the positive x-direction given by (PI2(Ax, ~b/2VH) = q)°2 + (2rt/dpo)BfAx
However, note that at this moment, electrons will be flowing from side 2 to 1 carrying the maximum current Jc, so the electron under consideration will continue to move in the positive x-direction. However, if vx, the speed of the electrons carrying the system current was such that at t = qSo/2VH the distance travelled in the positive xdirection was 2Ax, then the electron under consideration
Bf2Ax = ~b0.
Note that as the electron under consideration moved from side 2 to side 1, the phase difference between the two sides changed by n/2 throughout the system; further as it moved the distance 2Ax along the side 1 the phase difference between the two sides changed by n/2 more. Now one can easily see that as it crosses over to side 2 and moves 2Ax distance in the x-direction along the edge-current to reach the point x = 0 where it originally started, the ~012 would change all together by 2n. Thus, the result (20) that the rectangular loop, 6(2Ax), performed by the electron encloses a flux quantum is consistent with the q~zz changing by 2n in this process and, thereby, ensuring single-valuedness of ~.
V. Srivastava / Physica B 212 H995) 299-304
6. Quantum interference
Mixing of newly entering electrons with those which are phase shifted by 2rt after traversing the loops results into quantum interference which should show up as a maxima-minima pattern in the system current. It is easy to notice in Fig. 1 that if the system accommodates an integral number of flux quanta, say n, and the length L of the system is such that n(2Ax) ~< L < (2n + 1)Ax then the net current in the y-direction will be + Jc, where as, if (2n + 1)Ax ~< L < 2(n + l)Ax, i.e., after enclosing n flux quanta there is space for more than one half loop then the net transverse current will be zero. So one expects to observe a modulation in the net system current as one flux quantum is added to the s y s t e m - i t should decrease by J¢ when nq~oare trapped, then come back to its original value as B is increased, and again decrease by J¢ when (n + 1) flux quanta are exactly accommodated. This is similar to the single-slit interference pattern - here caused by a steady change in the magnetic field. As Anderson I-7] observed in the case of Josephson tunnel junctions and is equally applicable here, the change in magnetic field gives a linear variation of the phase across the sample (see Eq. (12)) in the same way that changing the angle of observation across a slit gives a linear change in phase across the slit. The periodic modulations of the system current should be reflected in the form of modulations of the same periodicity in the magnetoresistance and the Hall resistance. We have suggested  that the periodic oscillations observed by Mottahedeh et al. I-9] were induced due to the effects discussed here. Mottahedeh et al.  studied QHE in a narrow (width ~ 0.4 Bm) 2DEG of length 100 p.m over a temperature range of 64-765 mK and observed neat periodic oscillations in the magnetoresistance in the regions of plateaus i = 2 and 4. The period of oscillation was 0.065 T for i = 2 and was reduced by a factor of two to 0.033 T for i=4. In the present picture suppose a flux cp is penetrating the 6 x L strip in between the two edge-currents, and B(6 x L) = tp = nqgo,
This should correspond to the observed period of oscillation as discussed earlier. The plateau i = 2 at B -~ 13 T (correspondingly Vn = 13 x 10 -6 V) we get AB "-- 0.01 T. Its comparison with the observed value of 0.065 T can be considered good in view of the fact, as elaborated below, that we are unable to calculate vr (and, therefore, fi) properly due to the inadquate information available. It is, nevertheless, reassuring that the filling fraction dependence of AB comes out correctly from Eq. (23) the Vh=4 will be half of Vh=2, so (AB) i=* will be half of (AB)i=2. Some of the numerical simulations being tried at the University of Lancaster (UK) and Toho University (Japan) are also exhibiting periodic oscillations in the Hall resistance in narrow quantum Hall systems. These attempts are in preliminary stages and it will not be appropriate to give more details here.
7. Bottle neck
We saw in Eqs. (17) and (23) that vr, the velocity of phase-driven electrons, is required to be known for calculating 6, the width of the region over which the AC flows between the two edge-currents. This information is required to calculate AB accurately. We can write Eq. (20) in terms of vx and vr as vxv r = 16 V2/(Bdpo) = 50.36 x 103.
In spite of the right-hand side being known accurately, we cannot determine vr accurately since vx also cannot be ascertained correctly. Although Vx is fixed by the magnitude of the system current which is measurable accurately, yet to determine vx we need to know the difference of the number of electrons moving to the right and to the left, Inr - n l [ , in the two edge-currents respectively and this information is not available from the Mottahedeh experiment . All we know is that the system current is kept fixed at around 1 nA, and the carrier concentration is around 6 x 1015 m-2. Taking Inr - nil to be of order 107 m 1,we calculate vx as vx = l n A / I n r - nile ~ 103 ms -1.
Then, when exactly n loops of the kind discussed above (each enclosing one ~bo) fit into the gi x L strip. Now B is increased by AB such that one more flux quantum is added into the 3 x L strip, so (B + AB)(3 x L) = q~ + q~o.
vr --- 50 m s - 1
This approximate value has been used in calculating the AB from Eq. (23).
(22) 8. Discussion
Consequently AB = ~bo/(3 x L) = 4 V n / ( v r L ) .
We have seen that in order to observe a Josephson like effect in the QHE regime we need to do merely
V. Srivastava / Physica B 212 (1995) 299-304
a conventional QHE experiment-the only special requirement being that the 2DEG should be so narrow that the edge-currents are near enough to be weakly coupled and the length is less than the inelastic mean free path to ensure long-range phase coherence. The transverse AC thus setup between the edge-currents will manifest itself in the form of periodic oscillations in the magneto- and the Hall-resistance. The AC can itself cause a magnetic field which may be detectible. Also, one can attempt to observe Shapiro steps  in the I V characteristics of the Hall device by adding an AC component to the Hall voltage and then varying its frequency to resonate with that of the phase-driven AC. As regards formation of periodic oscillation in the magneto- and the Halt-resistances we learned an important thing in Section 5 - to enable certain electrons to form the loops, the velocity of the electrons in the edgecurrents must be adjusted carefully by suitably adjusting the system current so that the electrons are able to traverse a particular distance (2Ax) in a prescribed time interval (At = q~o/4Vn). This does not appear to be a difficult condition to be achieved. We feel that this condition was met accidently in the Mottahedeh et al. experiment. For the width 6 we do not have a definite prescription to suggest as to what the width of the 2DEG should be so that the rigid strength of the coupling is achieved. The relation (24) shows that v:, and Vy are coupled, so that when vx is adjusted, vy is also altered and so 6 is altered. Thus the variation of the system current not only controls the distance to be travelled in the x-direction but also that which is required to be travelled in the ydirection. This may demand adjustments in the width of the 2DEG (through say electrostatic squeezing) to achieve an optimum coupling between the edge-currents. It is clear that the relation (24) can be very helpful in estimating the system size provided one can measure in the experiment either the 6 or In~ - nd precisely. Note that the natural requirement for the AC to flow is that 6 should be larger than the separation w between the edge-currents. The condition for the AC to be setup can be determined for Eq. (24) as (4VH/BI)Inr -- n~le >~ w.
What gives strength to the relation (24) and in turn to the central result (23) is the fact that the order-of-magnitude calculation of 6 done through Eq. (24) agrees well with its estimate obtained using energy time uncertainty relation. While Eq. (24) gives 6 = 2 x 10 -9 m for i = 2, the position uncertainty in the y-direction in our system comes out to be about 2.5 × 10 -9 m. Note that 6 should indeed be comparable with the uncertainty Ay since the possibility of leakage of electrons between the two edge-currents creates the position uncertainty which should be of the
order of the distance over which the electrons are being exchanged, i.e., the distance over which the 'flip-flop' is occurring between the two states, namely the edge-states.
9. Connection with quenching of the Hall effect We have also pointed out a possible connection between the quenching of the Hall effect reported by Roukes et al.  in quasi-lD samples and the present Josephson-like effect in the low-frequency limit . It was shown  that as soon as the magnetic field is turned on and VH begins to develop the phases at the two edges begin to slip with respect to each other and drive electrons from one edge to the o t h e r - interestingly and significantly the phase-driven current is opposite in direction to that driven by the Lorentz force. Upto a critical value of the field the two currents compensate each other exactly and the Hall voltage does not develop (this is the quenching of the Hall effect).
10. Conclusion In sum we have argued here that a Josephson-like effect is embedded in the quantum Hall effect set-up and reveals itself in a special, though simple, geometry. We already find some experimental support for it in the experiments of Ref. [9, 11]. Further experimentation is suggested to establish the effect conclusively. Important conditions are worked out for seeing the effect in the experiment.
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