Table Of ContentElectron Transport in a Multi-Channel One-Dimensional Conductor: Molybdenum
Selenide Nanowires
Latha Venkataraman, Yeon Suk Hong, and Philip Kim
Department of Physics, Columbia University, New York, New York 10027
(Dated: February 6, 2008)
6
0 We have measured electron transport in small bundles of identical conducting Molybdenum Se-
0
lenidenanowireswherethenumberofweaklyinteractingone-dimensionalchainsrangesfrom1-300.
2
The linear conductance and current in these nanowires exhibit a power-law dependence on tem-
n perature and bias voltage respectively. The exponents governing these power laws decrease as the
a number of conducting channels increase. These exponents can be related to the electron-electron
J interaction parameter for transport in multi-channel1-D systems with a few defects.
9
1 PACSnumbers: 72.15.Nj,73.23.Hk,73.90.f
]
l Interacting electrons in one-dimensional (1D) metals
l
a
constitute a Luttinger Liquid (LL)[1], in contrast to a
h
Fermi liquid (FL) in 3-dimensional (3D) metals. Trans-
-
s port properties of 1D conductors are strongly modified
e
as adding an electron to a 1D metal requires changing
m
the many-body state of its collective excitations. This
.
t results in vanishing electron tunneling density of states
a
m at low energy. Power-law dependent suppression in tun-
neling conductance has been observed in many systems,
-
d including fractional quantum Hall edge states [2], sin-
n gle and multi-walled carbon nanotubes [3, 4, 5], bun-
o
dlesofNbSe3 nanowires[6]andconductingpolymers[7].
c
A cross-over from a truly 1D Luttinger-liquid (LL) to
[
a 3D Fermi-liquid (FL) is expected as 1D conductors
1
are coupled together, increasing the number of weakly
v FIG.1: (coloronline)(a)Structuralmodelofa7-chainMoSe
4 interacting channels [8, 9]. This transition, however, nanowire along with thetriangular Mo3Se3 unitcell, (b)and
5 has not been observed in the above (quasi) 1-D systems (c) AFM height images of MoSe nanowires between two Au
4 due to the experimental difficulty in preparing identi- electrodes. The wire heights are 7.2 nm and 12.0 nm respec-
1 cal conducting quantum wires to form conductors with tively. Scale bar= 500 nm.
0
a few weakly interacting channels. In this letter, we re-
6
port temperature and bias dependent electric transport
0
/ measurementsonsmallbundlesofMolybdenumSelenide Typically, 35 nm thick Au electrodes with 5 nm Cr ad-
at (MoSe) nanowires [10, 11, 12], whose diameter ranges hesion layer separated by ∼ 1 µm were used to contact
m from1-15nm. Thesenanowires,whichconsistofbundles randomly deposited nanowires. Figs. 1(b) and (c) show
- of weakly interacting and electrically identical 1D MoSe atomic force microscope (AFM) images of typical de-
d molecular chains, show a power-law dependent tunnel- vices. The two-probe resistance of such a device, which
n
ing conductance. The exponentgoverningthe power-law ranged from ∼ 100 kΩ−100 MΩ at room temperature,
o
decreases as the bundle diameter increases, indicating was measured in a cryostat with a continuous flow of
c
: a transition from 1D to bulk transport with increasing helium. A degenerately doped silicon substrate, under-
v
number of conducting channels. neath a L = 300 nm thick silicon dioxide dielectric
i ox
X Crystalline bundles of MoSe chains are obtained from layer,servedas a back gate to modulate the charge den-
ar the dissolution of quasi-1D Li2Mo6Se6 crystals in polar sityinthenanowires. Oncetransportmeasurementswere
solvents. Single crystal Li2Mo6Se6 was prepared as de- complete,the wirediameter,D,wasdeterminedfroman
scribedpreviously[13]. X-raydiffractionanalysisshowed AFM height profile.
hexagonally close packed molecular MoSe chains with a Fig. 2 shows the conductance (G) normalized by its
lattice spacing a0 = 0.85 nm, separated by Li atoms room temperature value as a function of temperature
(Fig. 1(a)). Atomic scale bundles of MoSe nanowires (T) for a representative subset of the samples studied
were produced from ∼100 µM solutions of Li2Mo6Se6 in [20]. We applied a small bias voltage, (V ≪ kBT/e), to
anhydrousmethanol. The solutions werethen spun onto stay in the linear response regime for this measurement.
degenerately doped Si/SiO2 substrates with lithograph- Notably, the mesoscopic scale samples (D < 20 nm,
ically patterned electrodes in a Nitrogen atmosphere. L ∼ 1 µm) exhibit more than two orders of magnitude
2
> 1 µm
tance follows a power-law remarkably well. Another po-
100 101 10 K tential explanation is a non-metallic behavior associated
0 K 12.0 nm with a Peierls transition, which opens up an energy gap
G3010-1 nS)100 at the Fermi level. However, from the conduction mea-
G/ G ( surements onthe bulk quasi-1Dcrystals (D >1µm)and
e,
c also from previous scanning tunneling microscopy work
uctan10-2 10--10.2 -0.1 4 K on similar nanowires [12], we observe no evidence for a
nd Gate (V) gap opening at temperatures down to 5 K, consistent
o
C with band calculations [15]. Alternatively, a power law
e 10-3
v dependent tunneling conductance is predicted for tun-
elati neling into a Luttinger Liquid, for 1D Wigner Crystals
R 5.3 nm
10-4 7.3 nm [16], for a highly disordered systems where the electron
1.2 nm mean free path is comparable with the wire diameter
3.0 nm [17]. We can rule out a 1D Wigner Crystal model since
10-5 theCoulombinteractions,whicharescreenedbytheback
10 100
gate,arenotlong-ranged. Wealsoeliminatethescenario
Temperature (K)
for a strongly disordered system by estimating the elec-
tronmeanfreepath,l inthenanowire. Wecanestimate
FIG. 2: Relative conductance (G/G300K) versus T for six e
mesoscopic and two bulk MoSe wires labelled with the wire leindirectlyfromtheeffectivewirelength,Leff,obtained
diameter. Curves are offset vertically for clarity. The solid fromthe dependence ofthe Coulombchargingenergyon
lines are power-law fits to the data (G ∼ Tα). The dashed the gate voltage. Here, the charging energy E = e2/C,
c
line indicates the region where Coulomb blockade becomes where the wire capacitance C ≈ 2πǫL /ln(4L /D).
important in two wires (N and H). Inset: Gate dependence eff ox
For the nanowiredevice shown in the inset of Fig. 2, the
of conductance(G) for the5.3 nm wire (N) from 4K to10 K
estimated charging energy E is 5-10 meV from the con-
at 1K intervals. c
ductance map in V and V (not shown), from which we
g
obtained L ∼ 0.3−0.6 µm. With this estimate, we
eff
conductancedecreasewithdecreasingtemperatureinthe ruleoutthepossibilityofhavingahighlydisorderedsys-
measured temperature range, unlike the samples in the tem. Moreover, the fact that the measured resistance is
bulk limit (D > 1 µm, L ∼ 100 µm), which exhibit a larger than ∼100 kΩ and is not directly correlated with
bulkmetallicbehavior,astheconductanceincreaseswith the wire diameter indicates that transport is dominated
decreasingtemperature,i.e. energy. Apower-lawdepen- by tunnelling. Thus a model concerning tunnelling into
dence, G ∼ Tα, is evident in these mesoscopic samples, a relatively clean LL is a more likely description of the
where the exponent α can be readily extracted from the observed transport phenomena [22].
slope of the least-squaresfit line in the double logarithm Further support for the LL-like transportin the MoSe
plot. For most of the samples, G(T) can be expressed nanowires can be found in the bias dependence of the
by a single α within the experimentally accessible con- conductance in the non-linear response regime. Accord-
ductance range [21]. However, for some wires with a ing to the LL model in a tunneling regime [1], the bias
relatively high conductance (> 1 µS) at room tempera- voltage dependent transport current, I(V), has a tran-
ture (Fig.2 N andH), anabruptchangeinthe exponent sition between an Ohmic behavior, i.e. I ∝ V in the
at low temperatures was observed. In this low tempera- low bias regime (V ≪ k T/e), and a power law behav-
B
tureregime,theconductancevariedwiththegatevoltage ior with an exponent β, i.e., I ∝ Vβ+1 in the high bias
(Fig. 2 inset) and the exponent depended on the applied regime (V ≫ k T/e). The inset of Fig. 3 shows typical
B
gatevoltage,V ,duetoCoulombchargingeffects. Above I(V) data measured in a mesoscopic wire with the ap-
g
this Coulomb charging temperature, a general trend of plied bias voltage ranging over more than three orders
decreasing α with increasing D is found for all meso- ofmagnitude at differenttemperatures. A transitionbe-
scopic samples studied. This trend will be discussed fur- tween Ohmic and power-law behavior is observed as V
ther later in the paper. increases.
We now consider severalpossible explanations for see- Interestingly, we also found that the exponent β de-
ing a decreasing conductance with decreasing tempera- pended strongly on D, as can be seen in Table I, where
ture. For example, one expects ln(G) ∝ −1/T for bar- welistα,βandDfor13samples. Ingeneral,wefindthat
rier activated transport. For highly defective wires, one α decreases monotonically as D increases. Based on the
expects ln(G) ∝ −1/Tδ due to variable range hopping relationbetween αand β, we cancategorizeour samples
between localized states in the wire, where δ can range into two distinct groups: group (I) where α ≈ 2β; and
from1/4fora3Dwireto1/2fora1Dwire[14]. However, group(II)whereα≈β. Inourexperiments,themajority
neitherofthesemodelsfitourdataas,forallmesoscopic ofsamples (10 outof 13in Table I) belongs to group(I),
scalemeasuredirrespectiveofwirediameter,theconduc- while only a few samples (3 out of 13 in Table I) belong
3
TABLEI:Measuredexponentsαandβdeterminedfromthetemperatureandbiasdependentconductancemeasurement,along
with the wire diameter (D) as determined by AFM and the number of channels including spin (N) calculated from D. Wires
indicated byasterisk (*) haveα≈β but for all other wires, α≃2β.
Wire W1 W2* W3 W4 W5 W6 W7 W8* W9 W10 W11 W12* W13
α 6.6 5.2 4.3 3.95 2.33 1.40 2.34 1.1 1.55 1.95 1.2 0.94 0.61
β 3.0 4.9 2.1 1.9 1.0 0.72 1.2 1.0 0.8 1.09 0.6 0.90 0.32
D 0.8±0.5 2.1±0.3 3.0±0.3 3.5±0.2 5.0±0.5 5.3±1.0 6.1±0.5 7.2±0.5 7.3±1.4 7.4±1.5 10.3±0.4 12.0±0.7 15.7±1.1
N 2 12 22 30 62 70 94 130 134 138 268 362 620
to group (II) [23]. high bias limit the FL-LL junctions become more resis-
For a clean LL without defects, these two exponents tive than the LL-LL junctions since αLL−LL >αFL−LL,
are expected to be identical, i.e., α = β [1], since they and thus I ∝ VαFL−LL+1. Therefore the exponents ob-
arecharacteristicofasinglejunctionbetweenFLandLL. tained from temperature and bias dependent data are
Thedeviationfromthismodelfoundingroup(I)samples expected to have a relation α = 2β for wires with a few
canbe explainedwithinthe LL modelwithafew defects defects that break them into multiple LL dots, as ob-
asdescribedbelow. Strongdefectsinthenanowiresbreak servedinourgroup(I)samples[24]. Thisargumentdoes
the conducting channels into a few LL dots connected in not hold however if there are no defects in the wire or
series between the electrodes. In this multiple LL dot inthe extreme ofstrongdefects that dominatetransport
scheme, the wires have two kinds of tunnel junctions; (i) within the experimentally accessible range of T and V.
junctions between the electrode and wire, constituting a For suchsamples,α≈β, which is the relationwe find in
FL to the end of LL junction; and (ii) junctions between group (II) samples.
two wire segments, constituting an LL to LL junction. The power-law behaviors in T and V allow us to scale
Insuchawire,the tunneling probabilitycanbespecified I(V,T)into asingle curve[3, 18]. Consideringthe above
by two distinct exponents αLL−LL and αFL−LL, where arguments,we can modify the scaling formula of a clean
αLL−LL = 2αFL−LL holds [3]. If the linear response LL transport model to include the two exponents α and
resistances of the FL-LL and LL-LL junctions at room β as:
temperature are of similar orders of magnitude, we ex-
pectG∝TαLL−LL inthelowtemperaturelimit,sincethe γeV β γeV 2
LL-LL junctions become most resistive. However,in the I =I0Tα+1sinh Γ 1+ +ı (1)
(cid:18)2k T(cid:19)(cid:12) (cid:18) 2 2k T(cid:19)(cid:12)
B (cid:12) B (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where I0 and γ are constants independent of T and V.
10-6 Physically, γ represent the ratio between the voltage
acrossadominantlyresistivejunctionathighbiastothe
C applied bias voltage[3]. As shownin Fig. 3, the seriesof
10-8urre I(V) curves measured at different temperatures for the
1 0-10nt (A) sdaemscerisbaemdpblyeEcoql.la1poseverremthaerkeanbtilryewmeellaosunrteodatseimngpleercauturvree
u.) 35K range by plotting I/Tα+1 against γeV/kBT with only
α1+ (a. 0.01 0.1 1 10-12 579505KKK 0o.n2e5±fit0ti.n1,gimpaprlayminegtethraγt.thFeroeratrheeprdoabtaabilnyfFouigr.b3a,rrγieriss
T
I/ Bias (V) 105K ofapproximatelyequalresistanceoverwhichthe applied
125K
bias voltage is distributed.
140K
160K Finally, we now discuss the dependence of the expo-
180K nents on the wire diameter (D) in order to elucidate the
200K
Fit transition from a few channel 1D transport to the 3D
transport limit. For this purpose, we focus on the sam-
0.001 0.01 0.1γeV/kT 1 10 100 ples with α = αLL−LL (i.e. group (I) samples and the
group (II) samples with γ ≈ 1). In Fig. 4, we show the
measured α plotted against D. Since N ∝D2, where N
FIG. 3: (color) Inset: Wire W3 I-V data taken at differ-
is the number of channels in the wire including the spin
ent temperatures between 35 K and 200 K. All curves show
a change from linear response to power-law dependence at a degree of freedom [25], the observed rapid decrease of α
temperaturedependentbiasvoltage. Mainpanel: I/Tα+1de- forlargediameternanowirebundlesindicatesacrossover
termined from I-V dataplotted against γeV/kBT with Eq.1 from 1D behavior to 3D transport (α ≈ 0). Employing
fit to the data. The measured exponents are α = 4.3 and the electron-electron interaction parameter for a single
β =2.1 (∼α/2) and thefitting parameter γ is 0.25±0.1.
chain (N=2), g, the end tunneling exponent αLL−LL for
4
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stantofsilicondioxide. Fromthe fitinFig.4,wededuce [20] The mesoscopic wires were susceptible to structural de-
that v = 3×104m/s. We note here that this value is formations locally at the wire/electrode junctions. Due
F
smaller than the value obtained from recent band calcu- totheinvasivenessoftheelectrodes,multi-terminalmea-
lation(4×105m/s)[15],indicatingthatastaticscreening surements were not possible.
[21] SinceGdecreasesrapidlyasT decreasesinthenanowire
picture consideredin this model might be too simplistic.
samples,mostofourG(T)datawerelimitedatlowT by
Further theoretical considerations including the effect of our current sensitivity (10−13A).
impurities and inter-chain hopping are needed to eluci- [22] We note that the environmental Coulomb blockade
date strongly interacting electrons in these 1D channels. (ECB) model (see for example, Ingold and Nazarov,
We thank C. M. Lieber, Y. Oreg,A. Millis, I. Aleiner, cond-mat/0508728) mightbeapplied toourexperiment.
B. Altshuler and R. Egger useful discussions. This work However, its validity is limited to the large N limit [8],
where the LL and ECB models do not offer any observ-
was supported by NSF Award Number CHE-0117752,
able differences in our experimental setup and we thus
by the New York State Office of Science, Technology,
consider only the LL model in this paper.
and Academic Research(NYSTAR). This work used the
[23] Group (II) samples can be sub-divided further into a
sharedexperimentalfacilitiessupportedbyMRSECPro- group with notable defects such as the one shown in
gram of the NSF (DMR-02-13574). Y.S.H acknowledges Fig. 1(b) (γ ≈1) and a group without strong defects
supportfrom the KoreanScience and EngineeringFoun- (γ ≈1/2), where the parameter γ is definedin Eq. 1.
dation. P.K. acknowledges support from NSF CAREER [24] For temperatures below 300 K, the deviation of the
(DMR-0349232)and DARPA (N00014-04-1-0591). measured α from αLL−LL is less than 10% as long
as RLead−Wire/RDefect is between ∼ 0.1 − 10 where
RLead−Wire is the total resistance between the wire and
theAu lead and R isthetotal resistance of all de-
Defect
fects. A similar tolerance range for RLead−Wire/RDefect
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0
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397, 598 (1999).
