Analogy Between Periodically Driven Josephson ac Effect and Quantum Hall Effect

4 Pages·0.12 MB·English

by  T. C. Wang

Analogy Between Periodically Driven Josephson ac Effect and Quantum Hall Effect
T.C. Wang
Department of Electrophysics, National Chiao-Tung University
Hsinchu, Taiwan
8 The hydrodynamical phase-slippage geometry and a semi-classical Josephson-RCSJ formulism are
9 combinedtomakeananalogy betweentheperiodically drivenJosephsonaceffectandthequantum
9 Hall effect. Both of these two macroscopic quantum effects are postulated to be originated from
1 the one-dimensional linear vortex-electron relative motion. The mechanisms of the quantized Hall
n resistance of IQHE and FQHE are unified under the phase-locking dynamics of a Josephson-like
a oscillation which is periodically driven by the electron wave of the coherent Hall current, with the
J time-averaged frequency ratio playing therole of thefilling factor reciprocal.

6 Supporting experimental evidences for this scenario as well as necessary future investigations are
1 remarked.

] PACS numbers: 73.40.Hm, 74.50.+r,05.45.+b
l
l
a
h
- Whentheappliedcurrentislargerthanacriticalvalue, Serving the metrology as another macroscopic quan-
s
e theJosephsonjunctiondevelopsanaccurrentalongwith tum standard of a classical quantity, the transverse Hall
m avoltagedropacrossthebarrier. Thisisknownastheac resistanceR in the Quantum Halleffect (QHE) [9][10]
H
Josephsoneffect[1]. If the junction is further exposedto can be quantized in units of h/e2:
.
at a microwave irradiation with an angular frequency ωr.f,
m the current-voltage characteristic curve shows Shapiro 1 h
R = , (4)
voltage steps [2] [3]: H νe2
-
d
~ once the longitudinal Hall current simultaneously be-
n
V =α ω , (1)
o 2e r.f comesperfectly conducting. In(4)ν is anintegerforthe
c integer quantum Hall effect (IQHE) [11] or a fractional
[ where 2e denotes the electric charge of the Cooper pair, number for the fractional quantum Hall effect (FQHE)
V represents the voltage drop across the junction, the [12].
3
step index α can be an integer or a fractional number
v Interestingly,weseefromadimensionalcheckthatthe
5 and the Planck constant ℏ = h/2π. According to such quantizedHallresistancecanbeexpressedash/e2multi-
4 a voltage-frequency relation, the r.f-biased ac Josephson pliedbythefrequencyratioofaJosephson-likeoscillation
1 effect provides a metrological quantum standard of the to an electron wave:
0 voltage.
1 To realize the Shapiro steps, one starts from the VH (~/e)ωH ωH h
7 Josephson relations [1] [4] RH = I [= ef =(ω )(e2)], (5)
9 H e e
at/ V = ~ dϕ (2) whereVH andIH istheHallvoltageandtheHallcurrent
2e dt respectively, [...] denotes dimensional equalities, therein
m
ω implies the angular frequency of a presumable single
where ϕ represents the electronic phase difference across H
-
chargeJosephsonoscillationand f (=ω /2π) represents
d the junction, and e e
n the frequency of a coherent electronwave. In the follow-
o I =I sinϕ (3) ingwork,wewouldliketoanswerthe inspiringquestion:
J c
c Is it possible to connect these two effects in a unified
: where I is the Josephson supercurrent, I is the criti-
v J c framework which solves the IQHE and FQHE by a com-
cal current, and then solves the semi-classical equation
i mon dynamical theory?
X describing the current conservation across the junction.
Compared to the simplicity the semi-classical phase-
r A discrete time computing of the well-known,r-f biased, lockingpictureofferedindescribingboththeintegerand
a
ResistivelyandCapacitivelyShuntedJosephsonJunction
fractional Shapiro steps in a unified scheme, up to date
(RCSJ [5] [6]) model shows [7] the main features of the
quantum mechanical approaches from IQHE to FQHE
I-V curve which includes various typical properties of a
appeartobe morecomplicated. Without loosingthe ba-
(2+1)-dimensional nonlinear dynamical system such as
sic commonquantum-mechanicalfeatures ofthose treat-
deterministic chaos, quasiperiodic and the phase-locking
ments,wetaketheGaugeInvariancepictureforexample
behavior. The Shapiro step index α can be shown as
to review the geometry, based on which the boundary
an integral, or a fractional, winding number [8], namely,
condition for QHE was given. If a closed integral path
time averaged frequency ratio < ∆ϕ/∆t > /ω , in the
r.f dl is taken to be a loop of electron current flow, as
phase-locked stationary states of the dynamical system.
H
1
that suggested by Laughlin’s ribbon loop [13], the quan- words, for a stationary electronic background as that in
tization behavior of the charge carriers in IQHE can be the zero-field Josephson ac effect mentioned above, the
attributed to the single-electron gauge invariance condi- relativemotionofchargeandfluxisrealizedbythe trav-
tion elingvortexwhileinastaticmagneticbackgroundofthe
quantum Hall system, we assume it resulted from the
A·dl=nΦ0, (6) propagating charge, namely, the Hall current that trav-
I els through static flux quanta.
whereAisthevectorpotentialofthegaugefieldthread- From this common phase slippage scenario, one finds
ing the looping current, l implies the position of the cir- that the electron fluid flowing along the x direction in a
planar Hall bar lying on the x−y plane would build up,
culating electrons and Φ0 denotes the gauge flux quan-
along the y direction across the flux, a voltage drop V
tum h/e. This current loop geometry provided a peri- H
odic boundary condition by which the phase increment togetherwithatimeevolutionofthephasedifferenceϕ12,
which satisfy a Josephson-likevoltage-frequencyrelation
of the electrons during each cycle as well as crossing
from one edge of the ribbon to the other was quantum- ~
mechanically restricted to be 2π, and the factor ν of (4) V = ω , (8)
H JH
(cid:18)e(cid:19)
forIQHEwasshowntobethenumberofchargesinvolved
in the cycling motion. According to this imaginary ge-
ometry, the FQHE can be realized by the introduction where the ”Josephson-Hall frequency” ωJH = dϕ12/dt.
With the spin-polarized electron playing the role of the
of a fractional charge state [14] for an interactive elec-
superconducting Cooper pair, this equation of the quan-
tron fluid within the many-body theoretical framework.
tum Hall system can be regarded as a single-charged
On the other hand, as to be illustrated below, the vor-
version of the Josephson relation established in a self-
tex motion scenario figured by P.W. Anderson [15] for
organized Josephson-like extended junction with the
the ac Josephson effect provides another possible, and
”virtual electrodes” separated by a linear channel lined
even more realistic, geometric relation between the Hall
up along x direction with parallel quantum fluxes. Ac-
current and voltage drop to show the quantized Hall re-
cording to this picture, one first see that keeping the
sistance.
appliedHallcurrentunchanged,wecanincreasetheHall
In the phase-slippage picture, a quantization condi-
voltage by two ways. One is to increase the density of
tion subjected to the Josephson voltage relation (2) was
fluxes or the applied magnetic field, the other is to de-
adopted in almost a same mathematical form compared
creasethechargecarrierconcentrationorthedevicegate
with (6):
voltage. Since both of the two operations increase the
number of fluxes passed, and thus the phase difference
∇φ·ds=2nπ, (7) ϕ12 experienced, by each propagating carrier within a
I
certain period of time. Such a continuous variation of
onlythattheintegralpath dswasnotablytakenaround the voltage would correspond to the ac Josephson effect
anisolatedJosephsonvorteHxcoreasshowninFig.1(a)in- without external driving. However, in addition to the
stead of a trajectory of circulating current, herein φ de- current-induced phase evolution of the Josephson vor-
noted the electronic phase. It should be noted that the tex, the coherent Hall current IH = efe itself results in
crucial underlying physical difference between these two another frequency ωe = 2πfe of the traveling wave of
pictures is that the former provides a quantization cri- electrons, or holes, which is also observed at any fixed x
terion for the single electron particle surrounding a flux position and further provides a periodic driving for the
while the latter has shownby itself a quantizationof the quantum Hall oscillation at this position. The system
vortex existing in the electronic fluid. To recall a real thus turns into a periodically forced quantum oscillation
image of the relative motion between electron fluid and and the quantized Hall resistance is able to be explained
vortices, one considers a thin layer Josephson junction by the phase-locking behavior between the Josephson-
lyingonthex−z planewiththevoltagedropdeveloping like oscillation and the Hall current. An example with
along the y direction and the vortex moving along the the locked frequency ratio 2/3, or ν = 3/2, is explicitly
x direction, see Fig.1(b). If the electronic phase differ- illustrated in Fig.2.
ence between side 1 and side 2 of the junction barrier, Althoughweregardthepictorialargumenttoberather
ϕ12(=φ1−φ2), at a certain x position is measured, one straightforward,thecorrespondingtheoreticalformulism
finds that it gains 2π increment as each traveling vortex turns out to be apparently unconventional. Instead of
passes through that x position. An analogy of quantum traditionallyquantizingtheelectronmotionunderexter-
oscillationcanthusbeintroducedintothequantizedHall nal field, one follows the Josephson-RCSJ approach to
system if we simply view QHE as the same relative mo- construct a semi-classical model for the Josephson-Hall
tion between the flux quantum and the electric charge oscillation which is modulated by electron waves. The
fromanotherobservationpoint,i.e.,thestaticvortexref- phenomenologicalcurrentconservationlawiny direction
erenceframe,seeFig.1(c),insteadoffromthestaticelec- at a fixed x position should thus appear like:
tronreferenceframefortheJosephsonaceffect. Inother
2
~d2ϕJH 1 ~dϕJH a quantum theoretical point of view. Foreseeable re-
C + +I g(ϕ )=|I |f(ω t), (9)
e dt2 Re dt c JH e e search should first include a unified dynamical study of
thequantumsolitoninbothsystem. Amongotherquan-
where ϕ representsthe ”Josephson-Hallphase”which
JH tum mechanical features, a relativistic Lorentz Contrac-
is also the electronic phase difference ϕ12 across the flux
tion[21]intheQHEmaybediscoveredasthecharge-flux
channel, the capacitance C and the resistance R are
relative velocity approaches the sample light velocity.
macroscopic parameters, |I |f(ω t) represents the pe-
e e
In summary, adopting the hydrodynamical phase-
riodic driving provided by the coherent Hall current,
slippage geometry together with the semi-classical
I g(ϕ ) represents a nonlinear vortex current for which
c H
Josephson-RCSJ dynamical formulism, this letter pro-
a simple image can be obtained if we designate g(ϕ )=
H posesapictorialanalogybetweenthe periodicallydriven
sinϕ to be the y component of a coherent circularly
H
Josephson ac effect and the quantum Hall effect. These
orbiting current within the quantized vortex. The Hall
two macroscopic quantum effects are postulated to be
resistance
both originated from the linear vortex-electron relative
~
R = VH = e ωH =(ωH)h (10) motion, with the moving and standing quanta been al-
H I (cid:0) e(cid:1)f ω e2 tered from the flux of the former to the electron of the
H e e
latter. The quantized Hall resistance of both IQHE and
can thus be shownto be quantized by solving (9) to find FQHE is suggested to be described by a unified phase-
a locked time-averagedfrequency ratio < ∆ϕH >/ω . locking dynamics of a Josephson-like oscillation period-
∆t e
From this simple dynamical approach, it has been ically driven by the electron wave of the coherent Hall
shownthatthe quantumphase-lockingscenarioprovides current, with the time-averaged frequency ratio playing
a unified mechanism for IQHE and FQHE. Experimen- the role of the filling factor reciprocal.
tal supports of this semi-classical model can be found
in the reported chaotic phenomenon [16] and a remark-
ablebistabilitywithbothdissipativeandnon-dissipative ACKNOWLEDGMENTS
states[17]inthecurrent-inducedbreakdownoftheQHE,
these typical nonlinear dissipative dynamical features I thank Prof. Y.S. Gou, Prof. C.S. Chu, Prof. B.
strongly suggest a ”chaotic” point of view for physicists Rosenstein and Prof. C.C. Chi for stimulating and help-
to deal with the QHE breakdown problem. Also to be fuldiscussions. ThisworkispartlysupportedbytheNa-
noted,thestrikingdiscoveryofthenovel1/2stateinthe tional Science Council of R.O.C. under grant No. NSC
multilayer system [19][20] can be intuitively understood 87-2112-M009-017.
within the phase-locking scenario as that the extra lay-
ers provide additional coupling to tame the previously
un-locked 1/2 state.
Further study is supposed to be twofold: Experimen-
taljustificationandrefinementsoftheequationofmotion
(9),aswellasitscomparisontotheexistingJosephsonef-
fectcounterpartcandomuchhelpstotacklecrucialopen [1] B.D. Josephson, Phys.Lett., Vol. 1, 251 (1962).
questions of QHE. For example, the sample-dependent [2] S. Shapiro, Phys. Rev.Lett., Vol. 11, 80 (1963).
occurrenceofthe even-denominatorstatesis expectedto [3] V.N. Belykh et. al., Phys. Rev.B. Vol. 16, 4860 (1978).
find its dynamical explanation in the ”Arnol’d tongue” [4] R.P. Feynman, Lectures on Physics, Addison-Wesley
(1963).
pattern [18] of the parameter space. The current and
[5] W.C. Stewart, Appl. Phys. Lett., Vol. 12, No. 8, 277
voltage distribution, whether in the edge or the bulk
(1968).
region, should be treated as a synchronized Josephson
[6] D.E. McCumber, J. Appl. Phys., Vol. 39, No. 7, 3113
junction array issue, no matter for the integer or the
(1968).
fractional quantized Hall states. The Farey series or-
[7] For a review, see R.L. Kautz, Rep. Prog. Phys., Vol. 59,
der of appearance of the quantized plateaux and the
935 (1996).
phase transition behaviors between the plateaux should [8] T. Bohr et. al., Phys.Rev. A30, No. 4, 1970 (1984).
alsobe comparedbetweenthesetwosystemunderauni-
[9] ForareviewonIQHE,seeR.E.Prangeet.al.,TheQuan-
fied dynamical framework. Finally, the very question re-
tum Hall Effect, Springer-Verlag(1990).
mainedtobeansweredturnsouttobehighlyfundamen-
[10] For a review on FQHE, see T. Chakraborty et. al.,
tal, if not been taken as philosophical. As we followed The Quantum Hall Effects, 2nd edition, Springer-
the Josephson-RCSJmethod to alternativelypropose an Verlag(1995).
electron-perturbed semi-classicalequation of motion, we [11] K. von Klitzing et. al., Phys. Rev. Lett., Vol. 45, No. 6,
havebeenforcedonlytoexaminethisformulismasanew 494 (1980).
macroscopic quantization rule, or in a practical sense, a [12] D.C. Tsui, H.L. Stormer and A.C. Gpssard, Phys. Rev.
semi-classical dual quantum theory. In either way, we Lett., Vol.48, 1559 (1982).
are obliged to see further extension of this temporalsys- [13] R.B. Laughlin, Phys. Rev.B., Vol. 23, 5632 (1981).
tem to a spatial-temporal mathematical structure with
3
[14] For a summarised introduction, see P.W. Anderson,
Physics Today,Vol. 50. No. 10, 42 (1997).
[15] P.W. Anderson,Rev.Mod. Phys., Vol. 38, 298 (1966).
[16] G. Boella et. al., Phys. Rev. B., Vol. 50, No. 11, 7608
(1994).
[17] F.J. Ahlers et. al., Semiconductor Science and Technol-
ogy, Vol.8, No.12, 2062 (1993).
[18] See for example, J.W. Norris, Nonlinearity, Vol. 6, 1093
(1993).
[19] Y.W.Suenet.al.,Phys.Rev.Lett.,Vol.68,1379(1992).
[20] J.P. Eisenstein et. al., Phys. Rev. Lett., Vol. 68, 1383
(1992).
[21] See for the example of Josephson junction, A. Laub et.
al., Phys.Rev.Lett., Vol. 75, No. 7, 1372 (1995).
FIG.1. (a)Anisolated vortexwithequal-phaselines. (b)
Foranon-propagatingelectronbackgroundastheacJoseph-
son effect, the x direction relative motion between flux and
charge comes from the translational motion of the vortex.
Duringthetimetf −ti, thevortex travelsfrom x1 tox2 and
the phase difference between side 1 and side 2 ϕ12 experi-
ences a 2π increment. (c) For a static magnetic background
as the quantum Hall effect, the relative motion comes from
the propagation of electrons. For a corresponding example,
duringthetimetf −ti,theelectron fluidflowsfrom x2 tox1
andthephasedifferencebetween1and2isalsoincreased by
2π.
FIG.2. Aonedimensionalcoherentelectronwavepassing
throughquantizedfluxesmakesaperiodically drivenJoseph-
son-like oscillation. Take β = 3/2 locked state as an ex-
ample: During the time ∆t, electron e1 originally at x1
passes through two fluxes f1 and f2 and causes a 4π in-
crement for the phase difference between side 1 and side 2,
which can be measured at any certain x position includ-
ing x2. Meanwhile, there are three electron wave packets
e1,e2 and e3 denoted by the sine curve, passing through x2.
Thus a phase-locked electron-driven Josephson-like oscilla-
tion results in at x2 the quantization of the Hall resistance
RH = VH/IH = (~/e)(4π/∆t)/(3e/∆t) = (2/3)(h/e2). Also
can be seen in this figure, the phase-locking dynamics corre-
sponds to an elastic deformation of the flux chain, i.e., the
linedensityofthefluxescanbeself-adjusted tomaintain the
locked density ratio of electrons to fluxes.
4