Download Coherent transport through a quantum dot in a strong magnetic field *

Physica B 249—251 (1998) 388—390
Coherent transport through a quantum dot
in a strong magnetic field
Michael R. Geller*
Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA
Abstract
I discuss recent work on the phase of the transmission coefficient for tunneling through a quantum dot in the quantum
Hall effect regime. The effects of electron—electron interaction on this phase is investigated with the use of finite-size
bosonization techniques combined with perturbation theory resummation. New non-Fermi-liquid phenomena are
predicted in the fractional quantum Hall regime that may be used to distinguish experimentally between Tomonaga—Luttinger and Fermi liquids. ( 1998 Elsevier Science B.V. All rights reserved.
Keywords: Chiral Tomonaga-Luttinger liquid; Aharonov—Bohm effect; Fractional quantum Hall effect
1. Introduction
In a recent series of beautiful experiments,
Yacoby et al. [1], Buks et al. [2], and Schuster et al.
[3] have succeeded in measuring both the phase
and amplitude of the complex-valued transmission
coefficient t"DtDe*( for tunneling through a quantum dot. The phase was measured by inserting the
quantum dot into one arm of a mesoscopic interferometer ring and observing the shift in the Aharonov—Bohm (AB) magnetoconductance oscillations. The weak-field experiments have already
stimulated considerable theoretical interest [4—11].
The same interferometer in a strong magnetic
field is strikingly different than that in weak field
because of the formation of edge states: Without
* Fax: 706-542-2492; e-mail: [email protected].
some phase coherence in the source and drain contacts there will be no magnetoconductance oscillations at all because the electrons will travel from
source to drain without circling flux [12]. Inserting
a quantum point contact or a quantum dot into
one arm of the strong-field interferometer couples
the inner and outer edge states together in a phasecoherent fashion. Because the coupling to the inner
edge state is assumed to occur in one arm only,
electrons scattered to the inner edge state must
eventually return to the outer edge state of that
same arm. Therefore, the effect of any inserted
scatterers is to introduce an equivalent scatterer
with transmission coefficient t. Usually, t results
from the transmission through an inserted mesoscopic structure in parallel with the inner edge
state of the ring. Because of chirality, t in the
strong-field case is a pure phase e*(.
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 3 7 - 9
M.R. Geller / Physica B 249—251 (1998) 388—390
The purpose of this paper is to present a brief
summary of the rich physics of the strong-field interferometer. To study the effects of electron—electron
interaction in this mesoscopic system, I shall use
bosonization techniques recently developed for the
finite-size chiral Luttinger liquid (CLL) [13]. The
dynamics of edge states in the fractional quantum
Hall effect regime is governed by Wen’s action [14]
P P
b
1 L
S "
dx dq­ / ($i­ / #v­ / ), (1)
B 4pg
x B
q B
x B
0
0
where o "$­ / /2p is the charge density flucB
x B
tuation for right (#) or left (!) moving electrons,
v is the edge-state Fermi velocity, and g"1/q (with
q odd) is the bulk filling factor. Canonical quantization in momentum space is achieved by decomposing the chiral scalar field / into a nonzero-mode
B
contribution /1 satisfying periodic boundary conB
ditions and a zero-mode part /0 . Imposing periB
odic boundary conditions [15] on the bosonized
electron field t (x),(2pa)~1@2e*q(B(x)eB*qpx@L leads
B
to the requirement that the charge N ,:L dxo
0
B
B
be an integer multiple of g.
389
The Fourier transform G (x, u) is particularly
B
interesting: For the case º"0, it is simply related
to the Green’s function for noninteracting (q"1)
chiral electrons [16],
e1~q q~1
< (u!j*e).
(5)
G (x, u)"Gq/1(x, u) F
B
B
(q!1)!
j/1
Whereas in the q"1 case the propagator has poles
at all integer multiples of the noninteracting level
spacing *e, in the interacting case the first q!1
poles (above the Fermi energy) are removed. This
effect, which can be regarded as a remnant of the
Coulomb blockade for particles with short-range
interaction, is a consequence of the factor q in the
first term of the zero-mode Hamiltonian. At higher
frequencies or in the large ¸ limit where uA*e, the
additional factor becomes uq~1/(q!1)!eq~1, charF
acteristic of an infinite CLL. Upon turning on
º a true Coulomb blockade develops, with a gap
given by º#(q!1)*e. I shall show below that the
rich low-frequency structure of Eq. (5) is directly
observable in the strong-field interferometer.
2. Retarded Green’s function for the finite-size CLL
3. Transmission coefficient
The study of mesoscopic effects in the CLL requires a careful treatment of the zero-mode dynamics.
I shall make extensive use here of the retarded
electron propagator G (x, t),!iH(t)SMt (x, t),
B
B
t† (0)NT for the finite-size CLL. In the presence of
B
an AB flux U"uU (with U ,hc/e) and addi0
0
tional charging energy º, the grand-canonical
zero-mode Hamiltonian corresponding to (1) is
The energy-dependent transmission coefficient is
equal to the ratio of retarded propagators
H0 "1q*e(N Ggu)2#1ºN2 !kN ,
B 2
2
B
B
B
where *e,2pv/¸. I then obtain
(2)
/0 (x, t)"$2pN (xGvt)/¸!gs
B
B
B
#g(k$u *e)t!gºN t,
(3)
B
where [s ,N ]"i, and (at zero temperature)
B B
G (x, t)"$(* )q(pa)q~1
B
L
]H(t)eB*qp(xYvt)@Le*(kBr*e)tSeB2p*qN(xYvt)@Le~*UNtT
A
]Im
B
e~*Ut@2
.
sinqp(xGvt$ia)/¸
(4)
G(x , x , e)
2 1
t(e),
,
(6)
G (x , x , e)
"!3% 2 1
with G
referring to the bare interferometer,
"!3%
which is the appropriate generalization of the
Fisher—Lee result [17] to the case where x and
1
x lie in interacting regions. It can be shown that
2
Eq. (6) gives the actual phase shift of the AB oscillations that would be measured in an experiment.
I now consider the case of tunneling through
a quantum dot weakly coupled to the interferometer edge states, as shown in Fig. 1. The Euclidian action for the system is S"S #S #dS,
`
D
where S is an action of the form (1) for the inter`
ferometer edge state, taken to be right moving, and
S is that for the edge state in the quantum dot. In
D
this configuration Coulomb blockade effects are
important, so a charging energy º is added to the
zero-mode Hamiltonian for the quantum dot edge
390
M.R. Geller / Physica B 249—251 (1998) 388—390
The resulting general expression will be given elsewhere.
The non-Fermi-liquid nature of the transmission
coefficient t(e) manifests itself as follows: At a fixed
energy e, the phase shift / as a function of the
quantum dot chemical potential k or “plunger”
D
gate voltage is the same as in a Fermi liquid (q"1),
but the effective coupling constants, which determine the resonance width, depend on e. However,
the energy dependence of t(e) at fixed k which can
D
be probed by varying the temperature or bias voltage, is dramatically different than in the Fermi
liquid case.
Acknowledgements
Fig. 1. A quantum dot weakly connected with the interferometer edge states. The contacts (not shown) are assumed to
be Fermi liquids.
state. The weak coupling of the quantum dot to the
leads is described by (i"1, 2)
P
b
dS"+ dq[vC t (x ,q)tM (x ,q)#c.c.].
(7)
i ` i
D i
i 0
Here x and x are the two tunneling points con1
2
necting the quantum dot to the interferometer edge
state. Note that quasiparticle tunneling is not allowed in this configuration. To leading nontrivial
order perturbation theory yields
A
B
G (¸ , e)
t(e)" ` */ # + v2DC D2 G (0, e)G (¸ , e)
i
D
` */
G (a, e)
`
i
#v2C C*G (L, e)G (a, e)
1 2 D2
`
[G (¸ , e)]2
` */
#v2C*C
G (L, e),
(8)
1 2 G (a, e)
D2
`
where ¸ is the length of the inner edge state and
*/
¸ the circumference of the quantum dot edge state.
It can be shown that DtD"1 as expected. This simple
result (8) is valid away from the quantum dot resonances. On resonance, where the quantum dot
propagator G (x, u) diverges, it is possible to calcuD
late t(e) to all orders in perturbation theory [18].
It is a pleasure to thank Hiroshi Akera, Eyal
Buks, Jung Hoon Han, Jari Kinaret, Paul Lammert, Daniel Loss, Charles Marcus, Andy Sachrajda, Amir Yacoby, and Ulrich Zülicke for useful
discussions.
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[15] These boundary conditions are the appropriate ones for
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