Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires
We report on the photoresistance of a magnetic quantum wire obtained by applying a gradient of magnetic field to a two-dimensional electron gas. Electron transmission through the magnetic wire increases by an order of magnitude under microwave irradiation and exhibits frequency dependent magneto-o...
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| Zitieren: | Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires / Alain Nogaret, Jean-Claude Portal, H.E. Beere, D.A. Ritchie, Chris Phillips // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1081–1085. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-1175662017-05-25T03:03:35Z Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires Nogaret, Alain Portal, Jean-Claude Beere, H.E. Ritchie, D.A. Phillips, Chris Carbon nanotubes, quantum wires and Luttinger liquid We report on the photoresistance of a magnetic quantum wire obtained by applying a gradient of magnetic field to a two-dimensional electron gas. Electron transmission through the magnetic wire increases by an order of magnitude under microwave irradiation and exhibits frequency dependent magneto-oscillations as a function of the in plane magnetic field. Both results are fully consistent with microwave coupled Luttinger liquid edge channels which interfere at two pinning sites in the fashion of a Mach–Zehnder interferometer. 2008 Article Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires / Alain Nogaret, Jean-Claude Portal, H.E. Beere, D.A. Ritchie, Chris Phillips // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1081–1085. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 73.21.Hb http://dspace.nbuv.gov.ua/handle/123456789/117566 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Nogaret, Alain Portal, Jean-Claude Beere, H.E. Ritchie, D.A. Phillips, Chris Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires Физика низких температур |
| description |
We report on the photoresistance of a magnetic quantum wire obtained by applying a gradient of magnetic
field to a two-dimensional electron gas. Electron transmission through the magnetic wire increases by an
order of magnitude under microwave irradiation and exhibits frequency dependent magneto-oscillations as a
function of the in plane magnetic field. Both results are fully consistent with microwave coupled Luttinger
liquid edge channels which interfere at two pinning sites in the fashion of a Mach–Zehnder interferometer. |
| format |
Article |
| author |
Nogaret, Alain Portal, Jean-Claude Beere, H.E. Ritchie, D.A. Phillips, Chris |
| author_facet |
Nogaret, Alain Portal, Jean-Claude Beere, H.E. Ritchie, D.A. Phillips, Chris |
| author_sort |
Nogaret, Alain |
| title |
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires |
| title_short |
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires |
| title_full |
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires |
| title_fullStr |
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires |
| title_full_unstemmed |
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires |
| title_sort |
microwave induced forward scattering and luttinger liquid interferences in magnetically confined quantum wires |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Carbon nanotubes, quantum wires and Luttinger liquid |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117566 |
| citation_txt |
Microwave induced forward scattering and Luttinger
liquid interferences in magnetically
confined quantum wires
/ Alain Nogaret, Jean-Claude Portal, H.E. Beere, D.A. Ritchie, Chris Phillips
// Физика низких температур. — 2008. — Т. 34, № 10. — С. 1081–1085. — Бібліогр.: 29 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT nogaretalain microwaveinducedforwardscatteringandluttingerliquidinterferencesinmagneticallyconfinedquantumwires AT portaljeanclaude microwaveinducedforwardscatteringandluttingerliquidinterferencesinmagneticallyconfinedquantumwires AT beerehe microwaveinducedforwardscatteringandluttingerliquidinterferencesinmagneticallyconfinedquantumwires AT ritchieda microwaveinducedforwardscatteringandluttingerliquidinterferencesinmagneticallyconfinedquantumwires AT phillipschris microwaveinducedforwardscatteringandluttingerliquidinterferencesinmagneticallyconfinedquantumwires |
| first_indexed |
2025-07-08T12:28:39Z |
| last_indexed |
2025-07-08T12:28:39Z |
| _version_ |
1837081790956699648 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1081–1085
Microwave induced forward scattering and Luttinger
liquid interferences in magnetically
confined quantum wires
Alain Nogaret
Department of Physics, University of Bath, Bath BA2 7AY, UK
E-mail: A.R.Nogaret@bath.ac.uk
Jean-Claude Portal
High Magnetic Field Laboratory, 25 Avenue des Martyrs, 38042 Grenoble, France
Institut Universitaire de France and INSA, 31077 Toulouse, France
Harvey E. Beere and David A. Ritchie
Cavendish Laboratory, University of Cambridge, CB3 0HE Cambridge, UK
Chris Phillips
Department of Physics, Imperial College, London SW7 2AZ, UK
Received March 31, 2008
We report on the photoresistance of a magnetic quantum wire obtained by applying a gradient of magnetic
field to a two-dimensional electron gas. Electron transmission through the magnetic wire increases by an
order of magnitude under microwave irradiation and exhibits frequency dependent magneto-oscillations as a
function of the in plane magnetic field. Both results are fully consistent with microwave coupled Luttinger
liquid edge channels which interfere at two pinning sites in the fashion of a Mach–Zehnder interferometer.
PACS: 73.21.Hb Quantum wires.
Keywords: Luttinger liquid, magnetically confined quantum wires, correlated electrons, microwaves.
In one dimension, the electron gas condenses into a chain
of strongly interacting particles called the Luttinger liquid
(LL) [1]. This liquid supports charge density wave insta-
bilities which are totally reflected even by the weakest link
in the chain [2–4]. The recent observation of power law
behavior characteristic of the LL insulator, in the thermo-
activated [5–7] and tunnelling current [8,9], appears as a
fundamental obstacle to scaling down electronic devices to
the size of single mode metal wires [10–14]. Here, however,
we report the first observation of quantum interferences in a
multi-channel LL [15,16] coupled by a microwave field at
two pinning sites where electrons experience forward scat-
tering. Experimental data fit quantitatively the response of
the LL confined by a spatially varying magnetic field
[17,18] and yield the distance between pinning centres
(11 μm) as the only adjustment parameter. The enhan-
cement of quantum coherence by Coulomb interactions
elicits applications to THz detectors [19], quantum com-
putation [3,17,20,21] and provides new insight into
correlated electron systems [9,22–24].
Magnetically confined quantum wires (MCQW) were ob-
tained by fabricating dysprosium micro-strips (length 40 μm;
width 200 nm; height 150 nm) at the centre of narrow Hall
bars (width 1.6 μm; length 32 μm) made of a GaAs/AlGaAs
quantum well (density ns = ⋅4 0 1011. cm–2, mobility m =
= ⋅ ⋅0 31 10 6 2. ( )cm V s/ set 30 nm from the surface [25]. A
magnetic field B was applied in the plane to magnetize the
stripe perpendicular to its long axis as shown in Fig. 1 (inset).
This had the effect of exposing the two-dimensional electron
gas (2DEG) to a fringing magnetic field B xm( ) of amplitude
varying between 0 and ±0.9 T. The magnetic field B was
aligned in the plane to a high accuracy (< 0.8°) and had no
direct effect on the 2DEG. An optically transparent titanium
film (30 nm thick) capped the magnetic wire to screen re-
© Alain Nogaret, Jean-Claude Portal, Harvey E. Beere, David A. Ritchie, and Chris Phillips, 2008
sidual electrostatic potentials and to protect the dysprosium
from oxidation. Samples were cooled to 1.3 K. Microwaves
were applied via an over-moded circular waveguide termi-
nated by linear polarizers and a 45° mirror redirecting power
onto the sample (E ⊥ wire). The output power was varied
from 0 to 20 mW using a variable attenuator. The photo-
resistance was measured under quasi-dc conditions using a
small current excitation (100 nA) which gave a spectral width
(~0.1 mV) comparable to thermal broadening.
Figure 1 shows the magnetoresistance of a 24 μm long
MCQW with and without microwave irradiation. This
exhibits a drop from 7.1 to 0.6 kΩ when microwave
power increases from 0 to 20 mW. In comparison, the
resistance of the bare Hall bar is 0.18 kΩ which is con-
sistent with the resistance of a channel of nominal width
supporting 75 modes. A much smaller photo-resistance
was observed across shorter channel lengths with
typically ΔR/R = −2% across a 4 μm segment. Surpri-
singly, microwaves induce complex magneto-resistance
effects: the most notable features being two series of
resistance oscillations and the recovery of the resistance
level prior to irradiation when B > 9 T. The resistance
dependence on microwave power is shown in the inset to
Fig. 1. To first approximation, the resistance decreases
linearly for B = 0 and 7.5 T whereas at B = 15 T, the
curve is independent of microwave power.
Figure 2,a displays the evolution of the magnetore-
sistance at frequencies between f = 75 and 110 GHz. The
observed structure is an effect of the spatially varying
magnetic field B xm( ). Firstly, because the amplitude of
peaks depends on the direction of the field sweep, a be-
havior associated with magnetization reversal in the mic-
ro-bar, see inset to Fig. 2,a. Secondly, one verifies that
1082 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Alain Nogaret, Jean-Claude Portal, Harvey E. Beere, David A. Ritchie, and Chris Phillips
Dy
2 DEG
B
50 Ghz
0
0 10 20
0 T
4
Power, mW
7.5 T
8
15 T
20 mW
0 mW
8
6
4
2
0
90 GHz
1.3 K
–5 0 5 10 15
B, T
Bm
Fig. 1. Microwave induced forward scattering. Main panel: Re-
sistance of a MCQW (200 nm wide, 24 μm long) with and with-
out microwave irradiation. The resistance at B = 0 drops by over
an order of magnitude when the microwave power is increased
from 0 to 20 mW. Top inset: A MCQW forms in a 2DEG con-
fined laterally by the stray magnetic field B xm( ) emanating from
a microscopic bar magnet. The external magnetic field B is ap-
plied in the plane to increase the perpendicular magnetization
which in turn increases the depth of the magnetic quantum well.
The photo-resistance is measured via four-terminal phase-loc-
ked detection under microwave irradiation. Bottom inset: De-
pendence of the photo-resistance on microwave power at B = 0,
7.5 and 15 T. The 0 and 7.5 T traces show a linear resistance de-
crease. The 15 T curve is essentially flat — only a small incre-
ase in resistance due to microwave heating is detectable.
20 mW
1.3 K
80 GHz
15
10
a
e f
A
A B C
a
e
d
f
e
f
A B
90
5
0
100
110
a –0.8 0.8
B, T
R
,
ar
b
.
u
n
it
s
y
y
0
a
a
c
d
e
b c d
e f
A B C
f,
G
H
z
110
90
d c b a a b c d e f A B C
70
50
–5 0 5 10 15
b
B, T
Fig. 2. Frequency dependence. (a) The longitudinal resistance
displays two series of oscillations. The low-field series
(a,b,c,d...) has peaks which move linearly with f. The peaks in
the high-field series (A,B,C...) fluctuate about an average value
which is independent of frequency. Curves are vertically offset
by 3 kΩ. Inset: magnetization reversal of dysprosium as eviden-
ced by the longitudinal magnetoresistance. (b) Fan chart of peak
positions. The dot size indicates the amplitude of experimental
peaks. The blue lines (color online) are the theoretical fan fitted to
the low-field series. The arrows indicate the theoretical peak po-
sitions of the high-field series (see text).
bare Hall bars exhibit no magnetoresistance under similar
experimental conditions [26]. The fan diagram in Fig. 2,b
maps the position of individual peaks in the f –B plane.
One distinguishes two series of oscillations. The low-
magnetic field series has peaks labelled (a–f) whose posi-
tion vary linearly with f . Peak (a) moves to lower mag-
netic field whereas peaks (b–f) move to higher magnetic
field. Peaks (A, B, C) are distinguished from the first
series by being independent of microwave frequency. We
shall see that, when B > 9 T, the transverse magnetization
of the micro-bar is within 4% of being saturated. As a
result, tiny fluctuations in the magnetization easily pro-
duce very large variations in the B-values of peaks (A, B,
C) as observed in Fig. 2,b.
The modulation field B xm( ) induces a magnetic poten-
tialV x k k e/ A x / my y y( , ) [ ( ) ( )] *≡ +� �
2 2 2 which depends
on two parameters: the longitudinal momentum k y and the
potential vector A x d By m
x
( ) ( )= ∫ χ χ
0
where m* is the elect-
ron effective mass. At high B, the stripe magnetization is sa-
turated perpendicular to the stripe ( M M⊥ = sat ) which al-
lows B xm( ) to be calculated exactly as a function of the
known dimensions of the stripe and the saturation magne-
tization of dysprosium (μ 0M sat = 3.67 T). The correspon-
ding magnetic potential is shown in Fig. 3,a. One notable
feature is the quantum well at the centre of the channel
which defines the MCQW. This quantum well is deep
enough (25 meV) to confine the Fermi sea (14.3 meV) and
sufficiently narrow (~200 nm) to support quantized energy
levels separated by ~0.5 meV, corresponding to ~5 k TB
where k B is Boltzmann’s constant. The energy subbands of
the MCQW are plotted in Fig. 3,b after numerically solving
Schr�dinger’s equation in potential V . Looking at this dia-
gram, one immediately sees that the effect of the magnetic
potential is to segregate edge states carrying opposite currents
to different regions of space. However, unlike in the quantum
Hall effect where N skipping orbits follow opposite edges,
we have here 2N magnetic edge states at the centre which
propagate in the negative y direction and N free electron
states at each edge of the channel which propagate along
y > 0, see the inset to Fig. 3,b.
Our experiment suggests the formation of a Luttinger
liquid in the MCQW. Firstly the pinning of a Luttinger liquid
by impurities explains the anomalously large resistance of the
wire in the absence of irradiation. Ogata and Fukuyama [27]
have calculated the conductance of the dirty LL (G) relative to
the conductance of noninteracting electrons (G0) as a func-
tion of the channel length (L), the mean free path (I), the
Fermi temperature (TF ) and T v /k LL F B≡ � . Our MCQW has
L l/ = 6, T/TF = 8 10 3⋅ − and TL = 90 mK for which the
theory gives G/G0 0 05≈ . . The experimental ratio taken from
Fig. 1 is G /G( ) ( ) .20 0 0 08mW = . The good agreement
between theory and experiment leads us to surmise that
microwave-excited electron-hole pairs introduce attractive
interactions in the Luttinger chain. These interactions switch
on forward scattering at pinning sites thus restoring perfect
transmission as predicted by Kane and Fisher [2]. If the
microwave heating of the electrons is inhibited as a result of a
perturbation applied to the system, forward scattering will be
switched off and the resistance will return to its level in the
absence of irradiation. This is precisely what happens in a
high magnetic field. Increasing the magnetic field increases
the energy gaps between the confined electron states in the
MCQW to the point where they become too large to absorb a
microwave photon. The quantitative analysis done below sets
the absorption edge at 8 T at 90 GHz which corresponds well
to the observed onset of the resistance plateau in Fig. 1.
Forward scattering reveals itself in more dramatic fa-
shion in the magnetic field dependence of the resistance
which we now describe. MCQWs have chiral edge states
Microwave induced forward scattering and Luttinger liquid interferences in magnetically confined quantum wires
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1083
E
(m
e
V
)
–4 43210–1–2–3
k , 1/ly b
15
10
5
0
a
b
–6
–4
–2
0
2
4
6
8
10
k
, 1/l
y
b
–800
–400
0
400
800
0
x, nm
40
30
20
10
E
, m
eV
EF
E
,
m
eV
n
Fig. 3. Energy subbands of a magnetically confined quantum
wire. (a) Magnetic potential V x k y( , ) induced by the stray mag-
netic field B xm( ) when the bar magnet is magnetized to full
strength in the perpendicular direction: μ μ0 0M M⊥ = =sat 3 67. T.
The MCQW forms in the potential well at the centre of the chan-
nel. The Fermi sea is shown in dark. (b) Quantum energy subbands
E kn y( ) supported by potential V x k y( , ). Magnetic edge states —
bound to the MCQW — are found at the left of the shadowed
parabola. To the right, the bonding/antibonding pairs are virtual
subbands lying above the MCQW potential barrier. Further to the
right, the group velocity changes sign and follows the free electron
dispersion curve. Free electron edge states form at the edges of the
Hall bar where Bm vanishes. The length scale is l /eb ≡ � = 25.7
nm. Inset: edge states in magnetically modulated Hall bar.
which by definition circulate in only one direction. This
has the consequence that charge density excitations circu-
lating along parallel LL channels will interfere in the fa-
shion of a Mach–Zehnder interferometer if the channel
paths are made to split and recombine at locations where
the LLs interact. Such interferometer is shown in Fig. 3,a.
Two weak links pin the LLs at y and y d+ . In the absence
of microwaves, these pinning sites block transmission and
cause charge accumulation as reported above. However in
the presence of a microwave field tuned at the frequency of
the inter-mode electron transition, electric dipoles allow a
finite tunnelling probability through the pinning sites
[17,18]. Two hybridized charge density waves propagate
between y and y d+ . These have velocity given by
s
v v V v v
KlF uF lF uF
± =
+
+ ±
−
+
2 2
10 2
π�
, (1)
where vlF and vuF are the Fermi velocities of the lower and
the upper channel, V e /0
2 4≅ πε is the Coulomb interaction
and K V / v vlF uF= −0 �( ) is the effective coupling strength
between channels [17]. We have noted ε the dielectric
constant of GaAs and e the electron charge. When Bm is
maximum, one obtains K = 18. Since K will invariably
increase for smaller Bm, the MCQW belongs to the strong
coupling regime: K >> 1. In which case, the speed of hyb-
ridized charge density waves is independent of ( )v vlF uF−
hence independent of the magnetic field. These modes de-
cay as d −λ where λ ± ≡ ± +1 1 2K/ K . The mode λ clearly
dominates and is retained for fitting the peak positions.
Theory predicts that this mode will absorb microwave
power according to a sin Φ function [17] where Φ is the
phase factor given in the left hand side of Eq. (2). Maxima
of power absorption occur when
k k
f
s
d
K
K
nlF uF− −⎛
⎝
⎜
⎞
⎠
⎟ +
+
= +
+
2
2 1 2
2
2
π π π π , (2)
1084 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Alain Nogaret, Jean-Claude Portal, Harvey E. Beere, David A. Ritchie, and Chris Phillips
^
y y+d
VlF
b
a
c
d
VuF
1.0
0.5
0
–0.5
–1,0
–15 –10 –5 0 105 15
B, T
0 5 10 15
B, T
16
14
13
B, T
15105A0–5
50
70
90
f,
G
H
z
110
B
E
n
,
m
eV
2
1
3
ky = 0
15
EF
Dy wire
M
/M
⊥
sa
t
Fig. 4. Quantum interferences of chiral Luttinger channels in MCQW. (a) Resonant microwave absorption couples two Luttinger
channels and enhances forward scattering at weak links (e.g., impurities) located at y and y d+ . Electron transmission through
the interferometer depends on the difference in Fermi wavevectors and Fermi velocities of the lower and upper channels.
(b) Transverse magnetization curve of a dysprosium bar — the up/down sweeps correspond to the curve 1/curve 2. Theoretical
fit: M/M B Bsat tanh tanh= +0 6 8 0 4 0 16. ( ) . ( . ) (curve 3). (c) Quantum energy levels E k Bn y( , )= 0 calculated as a function of the ap-
plied magnetic field using the empirical magnetization curve. Long (short) arrows show the electron transitions allowed at
110 GHz (50 GHz). B (A) is the upper limit of microwave absorption at 110 GHz (50 GHz). (d) The theoretical fan chart shows
the microwave absorption frequencies of the Luttinger liquid as a function of the magnetic field. The absorption maxima are la-
belled n = 6, 7, 8, 9, 10 (blue curves) and n = 13, 14, 15 (red curves) (color online) as explained in the text. The dotted line
(A–B) is the absorption edge. The distance d = 11μm is the only adjustment parameter in the theory.
where n is an integer and k klF uF– is the difference of
Fermi wavevector of the lower and upper channels. The
k klF uF– term carries the dependence on B. It thus beco-
mes clear that Eq. (2) describes a fan diagram in the f –B
plane that has parallel branches, each indexed by n.
A quantitative fit of the experimental data requires kno-
wing the magnetization curve M F B⊥ = ( ). Figure 4,b shows
the magnetization curve measured by Hall magnetometry
together with the function used to interpolate it. The latter is
used to calculate the B-dependence of energy levels, shown in
Fig. 4,c, and the B-dependence of k klF uF− . Inserting this
into Eq. (2) gives the slope of every branch in the fan without
any adjustment parameter. In contrast, the horizontal spacing
between two consecutive branches, n and n +1, depends on
distance d which we can extract from a fit to the data. The
best fit to Fig. 2,b gives d = ±( . )11 0 5 μm (blue lines). We
thus conclude that Coulomb interactions extend the range of
quantum coherence to a distance at least 3 times the electron
mean free path. The magnetic edge channel picture also
explains why the fan only appears at low B. The energy gaps
between MCQW subbands increase with B up to a point
where interchannel transitions are forbidden by energy
conservation. Figure 4,c predicts the absorption edge to occur
at 8.0 T (4.5 T) at 110 GHz (50 GHz) which fits nicely with
the upper boundary of the experimental fan in Fig. 2,b.
We finally address the origin of the high-magnetic
field series. This series is clearly nonresonant (as inde-
pendent of f ) and may be ascribed to charge density
waves excited indirectly, perhaps in the contacts. Setting
f = 0 in Eq. (2) allows to calculate the maxima of micro-
wave absorption as a function of n. The resulting theore-
tical positions are indicated by the arrows in Fig. 2,b
which are in good agreement with the data. We sum-
marize our findings by mapping the theoretical peak po-
sitions in Fig. 4,d. These replicate the major features of
the experiment. Our results demonstrate the enhancement
of quantum coherence by Coulomb interactions and inci-
dentally the quantization of energy in a spatially varying
magnetic field which is increasingly relevant to the deve-
lopment of carbon electronics [28,29].
Acknowledgments
We thank V.I. Fal’ko for stimulating discussions. The
support of the EPSRC(UK) EP/E002390 and the Trans-
national Access – Specific Support Action Program –
Contract RITA-CT-2003-505474 of the European Com-
mission is gratefully acknowledged.
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