Anomalous Density of States of a Luttinger Liquid in Contact with A
Anomalous density of states of a Luttinger liquid in contact
with a sup erconductor
1 2 3 1;4
C. Winkelholz , Rosario Fazio , F.W.J. Hekking , and Gerd Schon
1
Institut fur Theoretische Festkorperphysik, Universtitat Karlsruhe, 76128 Karlsruhe, Germany
2
Istituto di Fisica, Universit a di Catania, viale A. Doria 6, 95129 Catania, Italy
3
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
4
Department of Technical Physics, Helsinki University of Technology, 02150 Espoo, Finland
Abstract
We study the frequency and space dep endence of the lo cal tunneling density
of states of a Luttinger liquid LL which is connected to a sup erconduc-
tor. This coupling strongly mo di es the single-particle prop erties of the LL.
It signi cantly enhances the density of states near the Fermi level, whereas
this quantityvanishes as a p ower law for an isolated LL. The enhancement
is due to the interplaybetween electron-electron interactions and multiple
back-scattering pro cesses of low-energy electrons at the interface b etween the
LL and the sup erconductor. This anomalous b ehavior extends over large dis-
tances from the interface and may b e detected by coupling normal prob es to
the system.
Typ eset using REVT X
E 1
Transp ort in low-dimensional structures is strongly in uenced by electron-electron in-
teractions. A paradigm mo del to describ e interactions in one-dimensional systems is the
Luttinger liquid LL . The low-lying excitations of the electron system consist of charge and
spin waves, rather than quasiparticles [1,2]. As a consequence, the presence of a barrier
in the liquid leads to p erfectly re ecting for repulsiveinteractions or transmitting for
attractiveinteractions b ehavior at low energies [3].
One of the most striking characteristic prop erties of Luttinger liquids is the b ehavior of
the density of states DOS close to the Fermi energy. Contrary to Fermi liquids, whose
quasiparticle residue is nite, LLs have a DOS whichvanishes near the Fermi energy as a
power law,
g +4=g 4 =8
N ! ! : 1
The exp onent g dep ends on the strength of the electron-electron interaction: it is smaller
larger than two for repulsive attractive interactions. In the non-interacting case g =2,
and the DOS is constantasinFermi liquids [4].
Recently it b ecame p ossible to fabricate interfaces b etween a sup erconductor S and a
two-dimensional electron gas 2DEG [5]. An excess low-voltage conductance due to Andreev
scattering has b een observed in Nb-InGaAs junctions [6], as well as a sup ercurrent through a
2DEG in an InGaAs/InAlGaAs heterostructure with Nb contacts [7,8]. If the 2DEG is gated
to form a quantum wire, it should b e p ossible to study transp ort through Sup erconductor
- Luttinger Liquid S-LL interfaces. The Josephson current through a S-LL-S system has
b een calculated [9,10], as well as the I -V characteristics of a tunnel junction b etween a
sup erconductor and a chiral LL [11]. The latter can b e realized, e.g., in the fractional
quantum Hall regime.
The proximity e ect [12] mo di es the prop erties of a normal metal N in contact with
a sup erconductor. The leakage of Co op er pairs induces a non-vanishing pair amplitude in
N, de ned as F ~r =h ~r ~r i, where ~r is the annihilation op erator for an electron
" s
with spin s. The pair amplitude is a two-particle prop erty, related to the probability to nd
two time-reversed electrons at a p osition ~r . In a clean normal metal at zero temp erature,
F ~r decays as 1=r away from the N-S interface. In a LL with repulsiveinteraction in con-
tact with a sup erconductor, F ~r decays as 1=r , where >1 dep ends on the strength of
interaction [10]. These results hold as long as ~r lies within the temp erature-dep endent coher-
ence length =hv =k T from the interface to the sup erconductor. At larger distances,
N F B
F ~r decays exp onentially on the length scale . The reason for the decay of the pair
N
amplitude is that the two electrons lo ose their relative phase coherence over this distance.
Single electrons, however, lo ose phase coherence only at a much larger distance, namely the
phase-breaking length L . Indeed, recent exp eriments [8,13] have shown that interference
e ects due to single quasiparticles in N-S systems p ersist over distances much larger than
.We, therefore, exp ect quite generally a considerable in uence of sup erconductivityon
N
single particle prop erties over distances where the pair amplitude has already decayed.
In order to investigate these prop erties in a strongly interacting system we study the lo cal
single-particle density of states DOS of a LL in contact with a sup erconductor. We nd
that the lo cal DOS is substantially enhanced near the Fermi energy as compared to the p ower
law decay of an isolated LL cf. Eq. 1 . This result should b e contrasted with the b ehavior
of the pair amplitude a two-particle prop erty whichis suppressed in the interacting system. 2
The anomalous enhancement is a result of the interplaybetween the scattering of low-energy
electrons at the S-LL interface and the electron-electron interactions in the LL [14]. As for
the space dep endence, the DOS do es not decay in the same fashion as the pair amplitude
away from the S-LL interface. It remains enhanced up to distances of the order of the mean
free path, whichmaybemuch larger than for a clean quantum wire. Hence, the e ect
N
cannot simply b e explained in terms of a nite density of Co op er pairs in the LL. Both the
frequency and the space dep endence of the DOS can b e detected exp erimentally by coupling
normal metal tunneling prob es to the LL at some distance from the sup erconductor.
In the inset of Fig. 1 weschematically draw a 'clean' S-LL interface, consisting of a
LL in go o d contact with S. The shaded area indicates a tunnel junction b etween the LL
and a normal metal prob e. We will also study a LL of nite length connected to two
sup erconductors S-LL-S system . In such a system Andreev b ound states exist b elow the
gap [15]. Finally,we will comment on the case of a LL connected to S by means of a tunnel
barrier. The anomalous enhancement of the DOS is found in this case as well.
The Hamiltonian of a LL can b e written in b osonized form as h =1
"
Z
X
1 g 2
j
2 2
^
H = dx r ; 2 v r +
L j j j
2 2 g
j
j
where j = ; , and v = 2=g v are the renormalized interaction-dep endentFermi velo c-
j j F
ities. We restrict ourselves to repulsive, spin-indep endentinteractions; hence g < 2 and
g = 2. The Fermi eld op erators are decomp osed in terms of right- and left-moving Fermion
ik x ik x
F F
op erators and , resp ectively, = e + e , where k is the Fermi
+;s ;s s +;s ;s F
wavevector. The elds in turn can b e expressed through Boson op erators
;s
p
p
y
i [ x + x ]
s s
= e ; 3
0
;s
1 1
p p
where = + s and = + s . The density of electrons p er spin in the LL
s s
2 2
is = k =2 . Maslovetal. [10] recently develop ed a b osonization scheme to treat clean
0 F
S-LL interfaces. For a LL coupled to two sup erconductors at a distance L, they obtained
the following normal mo de expansion for the the elds
r r
X
x i g
y
^ ^
x = J + + sin qx b b ; 4
q ;q
;q
2 2L 2 2
q>0
r
X
i 1 g
0 y
^ ^
p
+ sin qx b + b ; 5 x =
q ;q
;q
2 2
q>0
s
r
X
2 x i
y
^ ^
x = b ; 6 M + sin qx b
;q q
;q
2 2L 2 g
q>0
s
X
1 i 2
0 y
^ ^
p
x = + sin qx b + b : 7
q ;q
;q
2 g
q>0
y
^
Here, b are Bose op erators and = expf q =2g where is a short range cut-o . The
q
j;q
phase di erence b etween the two sup erconductors is ; J and M describ e the top ological
0 0
excitations satisfying the constraint J + M = o dd. Finally, and are canonically
conjugate to J;M resp ectively. The lo cal density of states p er spin of the LL measured at 3
a distance x from the sup erconducting contact is obtained from the retarded one-electron
0 y 0
Green's function of the LL, G x; x ; t ihf x; t ; x ;0 gi t ,
R s
s
Z
1
1
i! t
N x; ! = Im dte G x; x; t : 8
R
1
We rst discuss the space and frequency dep endence of the DOS of a LL contacted at
x = 0 with a sup erconductor, which corresp onds to the limit L !1 in the mo de expansion
given by Eqs. 4 { 7 . In this case only the non-zero mo des q>0 contribute to the
y
lo cal DOS. The correlation function h x; t x; 0 i can b e evaluated using the b oson
s
s
representation Eq. 3 with the result
Y
2 2
j
j
2
+ 2x
y
h x; 0 x; t i =2
s 0
2 2
s
iv t
j
j=;
h i
j
2
; 9
i 2x+v t +i 2x v t
j j
at a distance x from the LL-S interface, where = g=16 1= 4g and = g=16 +
j j j j j
1= 4g . At small energies the DOS b ehaves as
j
g =4 1=2
N ! ! : 10
S LL
The exp onent of the DOS is negative g < 2 , which implies a strong enhancement at
low energies whereas in the absence of S the DOS of the LL vanishes at the Fermi energy.
The presence of the sup erconductor thus changes the prop erties of the Luttinger liquid in
a qualitativeway. Recently, Oreg and Finkel'stein [14] have found a similar enhancement
of the lo cal DOS of a LL in the presence of an impurity. They interpret their result as
a consequence of the interplaybetween the back-scattering induced by the impurity and
the repulsiveinteractions in the LL. A similar interplay exists in our system. Although
we consider a clean S-LL interface, backscattering is induced by the sup erconducting gap,
which re ects low-energy electrons either directly or via multiple Andreev pro cesses. The
enhanced DOS as a function of frequency, Eq. 10 , is schematically drawn in Fig. 1; for
comparison we also show the vanishing DOS in absence of the sup erconductor, Eq. 1 .
Atlow energies ! the enhancement of the DOS p ersists over large distances x ! v =!
from the interface. On the other hand, the induced pair amplitude in the LL, whichis
characteristic of the presence of the sup erconductor, decays as a p ower [10] of the distance
x. This profound di erence in the space dep endence demonstrates that the DOS provides
di erent information compared to the proximity e ect. The reason why the DOS do es not
approach the well-known b ehaviour of an Luttinger liquid far from the sup erconducting
contact is in part related to the fact that we are considering a clean wire. In this case the
states in the LL are extended and the DOS enhancement do es not dep end on x.
Wenow turn to the prop erties of the DOS for a S-LL-S system. The two sup erconductors
are separated by the distance L and are kept at a phase di erence . The latter can b e
achieved, e.g., byemb edding this junction in a SQUID. As we consider a LL of nite length,
the top ological excitations should b e taken into account; moreover the contribution from the
non-zero mo des consists of a discrete sum rather than a continuous integral over q states.
The correlator reads 4
Y
y
i v 1 = t=2L
F
h x; 0 x; t i = e D x; t : 11
;s j
;s
j =;
The exp onential prefactor originates from the top ological part; we used the fact that J =1
and M = 0 in the ground state for < <. The -dep endence is related to the phase-
dep endent shift of the Andreev levels see b elow . The contribution from the non-zero mo des
is given by
! !
2
j j
+2ix =L 2ix =L iv t =L
j
1 e 1 e 1 e
D x; t =
j 0
=L 2 =L
1 e 1 e
!
j
=L 2
1 e
: 12
[ i 2x+v t ] [ +i 2x v t ]
j j
1 e 1 e
As in the previous case of a single S-LL interface, the anomalous b ehavior of the DOS
at low ! extends over large distances, measured now relative to the p osition of one of the
interfaces , hence the phase-dep endent contribution to the DOS p ersists over distances much
larger than the Josephson coupling. If the interaction constant can b e written as a ratio of
twointegers g = m =n , we can express the DOS, using Eq. 11 , as
0 0
XX
N x; ! = a x [ ! E + !+E ] : 13
0 n ;n ;n
n
s;
Here a x are the Fourier co ecients of the function D x; t corresp onding to the energies
n
n
E = E 2 +1 v ;
;n F F
n 2Lk 2L
0 F
where E is the Fermi energy. The E are the energies of the Andreev levels [15] in the
F ;n
interacting quantum wire. The phase-di erence lifts the degeneracy for right- and left
moving electrons, giving rise to the Josephson e ect. In the noninteracting case g = 2 ,
all the -functions have the same weight and the lo cal DOS shows a p eak whenever the
frequency ! coincides with an Andreev level E . When the electron-electron interaction is
;n
switched on, the charge and spin part in Eq. 12 obtain di erent p erio dicities due to spin-
charge separation. As a consequence the co ecients a show a more structured b ehaviour.
n
In Fig. 2 the DOS is plotted for g = 1 as a function of the frequency and of the distance from
one of the two sup erconductors. For claritywe use a realistic, broadened version of the -
functions in Eq. 13 . We, further, xed the phase di erence between the sup erconductors
to zero. One clearly sees a strong enhancement of the DOS close to the interface. Away from
the sup erconductor the DOS remains enhanced, but the energy scale of the enhancementis
reduced to lower frequencies. The oscillatory contribution to the DOS is reminiscent of the
Friedel-oscillations, characterized by a p erio d 2k . In the general case 6= 0, the DOS for
F
the rightmoving electrons di ers from that of the left moving electrons due to the phase
factor in Eq. 11 . Although this leads to a more complicated dep endence of DOS on x and
! , the anomalous enhancement is still present.
So far we discussed the case in which the S-LL interface has a high transparency. Let
us shortly comment on the opp osite limit, in which the Luttinger liquid is connected to the
sup erconductor by a tunnel junction. In this case at low energies, we nd for the DOS close
g =2 1 + 1=2g g =8
to the junction N ! . Although the exp onent is di erent from the
S LL 5
one app earing in Eq. 10 , the DOS is clearly enhanced. Moreover, also in this case the
enhancement is found regardless of the distance from the junction.
In summary we considered the DOS of a Luttinger liquid in contact with a sup ercon-
ductor. We studied sp eci cally the cases of a single S-LL interface and a S-LL-S system.
Contrary to the well-known b ehavior in Luttinger liquids, the presence of the sup erconduct-
ing contact strongly enhances the lo cal DOS close to the Fermi energy, and this b ehavior
extends to large energy-dep endent distances from the interface. Our results can b e veri ed
exp erimentally [16], e.g., by means of the setup drawn in the inset of Fig. 1. We imagine
connecting the LL by means of a tunnel junction to a normal metal at a distance x from
the interface and measuring the I V characteristic of this junction. If there were no
sup erconductor the conductance of the normal metal-LL junction would go to zero as the
temp erature voltage, frequency is lowered. The presence of the sup erconductor leads to
an excess conductance at the junction.
ACKNOWLEDGMENTS
We thank C. Bruder for numerous imp ortant suggestions and W. Belzig and M. Fab-
rizio for useful discussions. The supp ort of the Deutsche Forschungsgemeinschaft, through
SFB 195, the Europ ean Community through contract ERB-CHBI-CT94-1764, and the
A.v.Humb oldt award of the Finish Academy of Sciences GS is gratefully acknowledged. 6
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[14] A similar anomalous enhancement of the lo cal DOS has b een recently found by Oreg
and Finkel'stein for a LL with a backscattering defect. Y. Oreg and A. Finkel'stein,
cond/mat 9601020 unpublished
[15] I.O. Kulik, Zh. Eksp. Teor. Fiz. 57, 1745 1969 [Sov. Phys. JETP 30, 944 1970 ].
[16] This setup has b een prop osed by D. Esteveet al. private communication to study the
DOS of a di usive conductor connected to a sup erconductor. A theoretical analysis has
b een p erformed by W. Belzig, C. Bruder, and G. Schon, to b e published in Phys. Rev.
B. 7
FIGURES
FIG. 1. Schematic dep endence of DOS on frequency for a pure LL dashed line and for a LL
connected to S solid line . Inset: Luttinger liquid, connected adiabatically to a sup erconductor.
The shaded area indicates a tunnel junction with a normal metal used to measure the DOS in the
LL at a distance x from the interface.
FIG. 2. The lo cal DOS for a S-LL-S system for particles of sp ecies p = , plotted as a function
of the frequency ! and the distance x from one of the S-LL interfaces. Wetookg = 1 repulsive
interactions ; the -functions of Eq. 13 have b een smeared, using p eaked Lorentzians with a width
6
of the order the level spacing. Furthermore, Lk =10 ;N = =E ; =0.
F L 0 F 8 Ν(ω)
Ν LL S S-LL N
9 x
Ν LL
ω 0.25 ω Np(x, )/NL 0.2
0.15 10
0.1
0
0.01 0.0004 0.0002 x/L 0.02 ω /EF