Mechanisms for electron transport in atomic-scale one-dimensional wires: soliton and polaron effects

7 Pages·0.64 MB·English

by  H. Ness

Mechanisms for electron transport in atomic-scale
one-dimensional wires: soliton and polaron effects
H. Ness ∗ and A.J. Fisher †
Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, U.K.
We study one-electron tunneling through atomic-scale one-dimensional wires in the presence of
coherent electron-phonon (e-ph) coupling. We use a full quantum model for the e-ph interaction
within the wire with open boundary conditions. We illustrate the mechanisms of transport in the
contextofmolecularwiressubject toboundaryconditionsimposingthepresenceofasoliton defect
in themolecule. Competition betweenpolarons and solitons inthecoherenttransport is examined.
The transport mechanisms proposed are generally applicable to other one-dimensional nanoscale
2 systems with strong e-ph coupling.
0
0 PACS numbers: 73.63.Nm, 73.40.Gk, 73.61.Ph
2

n
a I. INTRODUCTION of a Peierls transition. In this paper we address this
J question, in the context of molecular wires; specifically,

6 we consider finite-length poly-acetylene chains described
How will an electronic system conduct in the limit
1 by a fully quantum version of the Su–Schrieffer–Heeger
whereitbecomesentirelyone-dimensional? Thisbecame
(SSH) model [15].
1 an issue in the late 1980s, with the fabrication of semi-
v conductor structures that possessed only a single con- We have chosen to study molecular conductors here,
6 ducting“channel”,andwhichexhibitedthephenomenon rather than any of the other types, because electron-
7
of quantized conductance [1]. With nanoscale fabrica- lattice coupling in bulk samples is relatively well stud-
2
1 tion techniques, the class of effectively one-dimensional ied [15]. The Peierls transition manifests itself in an al-
0 systemshasexpandedtoinclude: atomic-scalestructures ternation of single and double bonds (i.e., dimerisation)
2 on surfaces produced by scanning probe lithography [2], along the molecule’s length, and bulk electron transport
0 self-assembly [3], or directed assembly [4]; break junc- is dominated by mobile intrinsic defects formed when
t/ tions[5,6];carbonnanotubes[7];andconjugatedorganic electrons or holes are injected into this dimerised struc-
a
m molecules[8]. Themeasurementofthetransportproper- ture. Forourpurposesthemostimportantdefectsarepo-
ties of individual examples of such systems is now possi- larons(whichcanbethoughtofasalocalreductioninthe
-
d ble; for example, conduction in individual molecules has dimerisationaroundaninjectedcharge)andchargedsoli-
n recentlybeenmeasuredbyscanningprobes[9,10]orbulk tons(topologicaldefects inthe bondlengthalternation).
o electrodes [8,11]; transport in nanotubes has been mea- Thesolitonhasanassociatedelectronicstateatmid-gap
c sured using lithographed electrodes [12]. Similar mea- and the dimerisation changes sign, passing through zero
:
v surements may also soon become possible for atomic- in the neighbourhood of the defect [15].
i
X scale wires on surfaces. In order to understand molecular charge transport on
r Many of these recent examples approach the ideal of nanometre lengthscales,we must develop a theory of co-
a a system whose atomic—as well as electronic—structure herent transport that accounts for polaron and soliton
is truly one-dimensional. As such they may be expected formation. We took the first steps recently when we
to exhibit enhanced effects of correlations during elec- showed that the tunneling of carriers through molecu-
tron transport. Electron-electron correlations have con- lar wires at low temperature is dramatically enhanced
sequences including the possible formation of Luttinger by the formation of virtual polarons [16]. Therefore,
liquids [13]; on the other hand, the effects produced by previous treatments of tunneling in molecules based on
electron-atomcorrelationscaninclude the Peierlstransi- elasticscatteringsignificantlyunderestimatetheconduc-
tion[14],whichopensagapattheFermienergyandren- tance [17]. However, the boundary conditions used in
dersone-dimensionalmetallicsystemssemiconducting. If ourpreviousworkpreventedsolitonsfromforminginthe
thesequasi-one-dimensionalsystemsaretohaveapplica- molecule,asthedimerisationwasconstrainedtobeequal
tionsinnanotechnology,itisvitaltounderstandwhether at both ends. In the present paper we study chains with
currents can pass through them, despite the possibility an odd number of monomers; the ground state of such a
∗ Present address: CEA-Saclay, DSM/DRECAM/SPCSI, Bˆat. 462, F-91191 Gif-sur-Yvette, France. E-mail: [email protected]
† E-mail: [email protected]
1
systempossessesasinglesolitoninthecentre[15,18]. We with a definite number of electrons in each part). The
are thus able to study for the first time the role of the single added carrier can be anywhere in the system (on
moving soliton, with its associated mid-gap state, and the left, the right, or in the molecule) and interacts with
the competition between polarons and solitons in coher- phonons only when inside the molecule. We only con-
ent transport. sider current-carryingstates in which a single electronis
added to the groundstate; we expect this assumptionto
hold since the interval between electron transmission is
II. PHYSICAL MODEL much greater than the transit time [16].
We now briefly outline the calculation technique.
The model for the molecular wires includes delocal- The transport problem is solved by mapping the many
izedπ-electronsinteractingwith quantumphonons. The body(one-electron/many-bosons)problemontoasingle-
electronic Hamiltonian is expressed in the basis of the electron problem with many scattering channels [20,16].
one-electron eigenstates (labelled by n,m) of the refer- Eachchannelrepresentsthedifferentscatteringprocesses
ence system. The e-ph coupling is linear in the phonon by which the electron might exchange energy with the
displacementsandinducestransitionsbetweenelectronic phonons. For an initial phonon distribution b m
λ
≡ { }
states. The molecule Hamiltonian is and an incoming electron from the left lead, the outgo-
ing channels in the left and right leads are associated
H =Xǫnc†ncn+X¯hωλa†λaλ+ X γλnm(a†λ+aλ)c†ncm ,withenergy-dependentreflectionrab(ǫ)andtransmission
n λ λ,n,m t (ǫ) coefficients respectively (a n is the phonon
ab λ
≡ { }
distributionafterscatteringineachoutgoingchannel). In
(1)
the leads, the scattering states take the asympotic form
where c†n (cn) creates (annihilates) an electron in the n- of propagating Bloch waves with amplitudes rab (reflec-
th electronic state with energy ǫ and a† (a ) creates tion) and tab (transmission). The wavevectors are given
n λ λ
bythedispersionrelationsineachchannel. Forexample,
(annihilates) a quantum ofenergyh¯ω in the eigenmode
λ
theenergyǫ oftheincomingelectronfromthe leftlead
of vibrationλ of the isolatedmolecule. The values ofǫ , in
n
is related to the wave vector kL by the tight-binding-
ωλ andγλnm are calculatedfromthe groundstate ofthe b
like dispersion relation ǫ = ǫ +2β coskL. For the
neutral molecule (described by the SSH model [15,16]) in L L b
finalenergyǫ ofthe electrontransmittedto the right,
containing an odd number N of monomers. The vibra- fin
one has: ǫ = ǫ +2β coskR. One can then project
tionaleigenmodesVλ andfrequenciesofthemoleculeare fin R R a
out the leads from the problem and work in the molecu-
calculated within the harmonic approximation [19].
lar wire subspace to solve for the value of the scattering
Tocalculatethetransportpropertiesthroughthewire,
state Ψ(E) . This state is obtained from propagating
the left and right ends of the molecule (atomic sites
| i
the source term s(E) (incoming electron from the left)
i = 1 and i = N) are connected to metallic leads | i
via the effective Green’s function G(E) defined in the
via hopping integrals v respectively. The leads are
L,R
molecular wire subspace: Ψ(E) = G(E)s(E) , where
modeled as one-dimensional semi-infinite tight-binding
| i | i
G is given by G(E) = [E H Σ (E) Σ (E)]−1, H
chains with on-site energy ǫL,R and inter-site hopping − − L − R
isthemolecularwireHamiltoniandefinedinEq. (1)and
matrix elements β (with no e-ph coupling). The
L,R
Σ (E) are complex potentials arising from embedding
scattering states Ψ for a single incoming charge car- L,R
| i themolecularwirespectrumintothecontinuumofstates
rier are expanded inside the molecule onto the eigen-
states n, n = c† (a†)nλ/√n ! 0 , of the non- associated with the leads. Overall energy is conserved,
interact|ing{eλ-p}hi systenmQ1λ. nλλ is the λocc|upiation number so ǫin and ǫfin are related by: E = ǫin+Pλmλ¯hωλ =
ǫ + n ¯hω 2.
of the phonon mode λ and 0 is the vacuum state (rep- fin Pλ λ λ
| i
resenting the neutral ground state of the whole system, The linear system Ψ =Gs is solved for a finite size
| i | i
1Asimilarexpansionisusedintheleads,c† isthenreplaced
n
by thecharge creation operator d† on site i.
i
2 In this paper, we consider the limit of low temperatures:
the initial phonon distribution is b ≡ {m } = {0} where all
λ
optic phonon modes are in the ground state (in this limit,
the injection energy equals the total energy ǫin = E =
ǫ + n ¯hω ). It is a very good approximation even at
fin Pλ λ λ
roomtemperatureforallopticmodesexceptthesolitontrans-
lation. For the soliton translation the condition kT ≪ ¯hω
restricts usto T ≪230K.
2
basis set by truncating the phonon subspace (i.e consid- considered for a molecule of length N =99. As an illus-
ering the lowest occupation numbers up to nmax in each trative example, we also show in Fig. 2 the dimerisation
occ
mode). Furthermore the electron is coupled inside the d for neutral isolated even- and odd-length molecules.
j
molecule to a finite number N of the most relevant Thedimerisationisobtained,intermsofatomicdisplace-
ph
phonon modes (see next section). A detailed analysis mentsu ,fromthestaggereddifferencebetweenadjacent
j
of the validity of such approximations can be found in bondlengths: d =( 1)j(u 2u +u ). ForevenN,
j j+1 j j−1
− −
Ref. [16]. From the solution Ψ = Gs , one can obtain theneutralmoleculeisperfectlydimerized: thedimerisa-
| i | i
the reflection r and transmission t coefficients for all tion d is constantin the middle of the chain despite the
ab ab c
the channels and hence the currents flowing through the end effects. For odd N, a soliton appears in the middle
wire. One can also calculate the expection values of any of the chain, acting as a domain wall separating the two
correlation functions between the electron and phonon domains of opposite sign of dimerisation.
degrees of freedom.
Now we turn on the lattice deformations induced by
We define the transmission probability T =
ab electron propagationin the (odd N) molecule connected
|tab(ǫin)|2βRsinkaR(ǫfin)/(βLsinkbL(ǫin)) 3. In our to the electrodes 5. The lattice deformations induced by
model, t 2 is proportional to i = N,aΨ 2 =
ab the tunneling electron are given by the correlation func-
| | |h | i|
v|heNlo,cait|Gy|osfi|t2haenidnctohmeinsoguerlceecttreornm[1is6]p.rTophoernt,iownoarlktiongthine tion δλ[i] between the electron density Pi = c†ici on site i
the real-space representation, the inelastic transmission and the displacement ∆λ = (aλ +a†λ)p¯h/2Mωλ of the
probability can be rewritten in the following usual form mode λ as δλ[i] =hPi†∆λPii/hPii. We also define the dis-
[21] : tortion on site j due to the displaced modes when the
electronis on site i as u[i] = δ[i]V (j). The dimerisa-
v2 v2 j Pλ λ λ
T (ǫ ,ǫ )=4 L sinkL(ǫ ) R sinkR(ǫ ) tion pattern d[i] is then calculated from u[i].
ab fin in β b in β a fin j j
L R
We plot on Fig. 3 the absolute value of the dimerisa-
i=N G (E)i=1 2 , (2)
×|h | ab | i| tiond[i] fromwhichtheconstantdimerisationd andthe
j c
where N G (E)1 isthematrixelementoftheGreen’s endeffectshavebeensubtracted(the correspondingpat-
ab
h | | i
functionGtakenbetweentheleftsidei=1andtheright tern for the isolated molecule is shown on Fig. 2). Such
sidei=N ofthemoleculeandthephononconfigurations a choice permits us to represent the defects with more
before (b) and after (a) scattering 4. contrast. For an injection energy E = 0 at mid-gap, the
corresponding dimerisation is shown on Fig. 3(a). The
bright feature around the middle of the chain represents
III. RESULTS
the soliton. In the absence of e-ph coupling, this would
correspondtochargeinjectioninresonancewiththeone-
First we describe the vibrational modes needed to ob- electron soliton level (see the transmission curve below).
tain the relevant distortions of the molecule. We have The soliton would then be immobile, its position fixed
shown that polaron formation inside the molecule af- in the middle of the chain and its width unchanged (i.e.
ter charge injection is due to the coupling to the long- straight vertical feature around j = 50). However due
wavelength optic modes [16]. For odd-number chains to the e-ph coupling, the soliton defect becomes mobile
where a soliton defect is present, we have to include in and its width varies while the electron progates. Similar
addition the modes responsible for the motion and the behaviouris observedforother injectionenergiesaround
deformationofthesoliton. Wethereforeincludethesoli- mid-gap. As E approaches the valence band edge the
ton translation (the ‘Goldstone mode’), the ‘amplitude mechanisms become different, as shown on Fig. 3(b).
mode’ related to the deformation of the soliton width, In parallel with the soliton delocalization and deforma-
andhigher-orderdeformationmodes(forexampletheso- tion, one observes the formation of a polaron. This is
called third mode) [15,22]. Figure 1 shows the phonons characterizedbytheextrabrightfeaturearoundthefirst
3 T contains as usual the ratio of the velocity in the out-
ab
going channelto that in theincoming channel.
4 ThefactorsvL2,R/βL,RsinkbL,,aR inEq.(2)arerelatedtothe
imaginary parts of thepotentials ΣL,R.
5 Calculations were performed for an electron coupled to
the N = 5 lowest frequencies modes shown in Fig. 1 and
ph
nmax =4. Calculationswerealsodoneforothersetsofparam-
occ
eters. The same qualitative physics remain when the results
are converged versusthebasis set size.
3
diagonal in Fig. 3(b). In principle, the (virtual) po- injectionenergies(stillinside the energyband-gapofthe
laroncorrespondstoalocalreductionofthedimerisation molecule), a virtual polaron can be formed. The trans-
aroundthe (tunneling) electron[16]. Howeverit appears port is then associated with more complicated mecha-
as an increase (positive number) since d[i] is plotted on nisms involving the interactionof both polaron and soli-
Fig. 3(b). Furthermore, the correspon|dijng| dimerisation ton. Howeverinmostcases,theeffectivereductionofthe
is not simply the superposition of the polaron and the band-gap due to polaron formation, the delocalization
soliton—the two defects interact strongly together. For and deformation of the soliton and the polaron-soliton
example, the dimerisation pattern for an electron at po- interaction increase the transmission in the tunneling
sition i 25 is characteristic of a merging of both the regime. Although we studied a model derived from or-
≈
polaronandsoliton. Suchmechanisms,leadingto strong ganic molecules, there is good reason to believe that the
latticedistortions,wouldbe expectedtoaffectthetrans- e-ph coupling in other one-dimensional nanoscale wires
missionthroughthemolecularwires—effectsthatwenow is likely to give rise to similar phenomena. In the case of
consider. carriers injected into dangling-bond lines on the Si(001)
We define an effective total transmission probability surface [2], polaron states which are in many respects
T(E) arising from the contribution of the different out- analogous to those in molecular wires are also formed
going channels as T(E) T (ǫ ,ǫ ) δ(ǫ + [23]. Specifically, we expect the present results to be
n ¯hω ǫ ) (see Fig.≡4)P. aWiatbhofuitne-ipnh coufpilning, validforanyatomic-scalewireinwhichthere isa degen-
Ptheλtrλansmλi−ssioinn curve presents the usual features: reso- eracy between two different values of some order param-
nances atenergiescorrespondingto the one-electronlev- eter (the dimerisation in the molecular case, the surface
elsofthe moleculewithalmostperfecttransmission,and dimer buckling in the dangling bond line) which are re-
strong suppression of the transmission in the band-gap latedbyadiscretesymmetryandstronglycoupledtothe
of the molecule for which propagation occurs by tunnel- electrons.
ing. Thesolitonresonance(atmid-gap)ismuchnarrower
than the other resonances. This is to be expected since
the corresponding one-electron (soliton) state is much ACKNOWLEDGMENTS
more localized than the other states. With e-ph inter-
action, we have already shown that a polaron can be
formedin the molecule andthat the solitonposition and We acknowledge financial support from the U.K. EP-
widtharestronglymodified. Suchmechanismsaffectthe SRC (Grant GR/M09193).
transmission in the following way: (i) the delocalization
anddeformationofthesolitonbroadenbutlowertheres-
onancepeakatmid-gap,(ii)forlargerinjectionenergies,
the formation of the polaron effectively reduces the ap-
parent band-gap, and a polaron resonance peak appears
(at E 0.42 eV for the N =99 wire) inside the original
≈ [1] van Wees B.J. et al., Phys. Rev. Lett. 60 (1988) 848;
gap. This effective band-gap reduction correlatively in-
Wharam D.A. et al.,J. Phys.C 21 (1988) L209.
creasesthe transmissioninthe tunneling regime(i.e. for
[2] Hitosugi T. et al.,Phys.Rev.Lett. 82 (1999) 4034.
injection energies in the gap). [3] Dong Z.C., Fujita D. and Nejoh H., Phys. Rev. B 63
(2001) 115402.
[4] LopinskiG.P.,WaynerD.D.M.andWolkowR.A.,Nature
IV. CONCLUSION 406 (2000) 48.
[5] Costa-Kr¨amer J.L. et al., Phys.Rev.B 55 (1997) 5416.
[6] Ohnishi H., Kondo Y., and Takayanagi K., Nature 395
The results presented above demonstrate the com- (1998) 780.
plexity of electronic transport through one-dimensional [7] Lemay S.G. et al., Nature412 (2001) 617.
atomic-scale wires. We have illustrated the mechanisms [8] Reed M.A. et al.,Science 278 (1997) 252.
ofelectrontransportinmolecularwireswhene-phinter- [9] Langlais V.J. et al., Phys.Rev.Lett. 83 (1999) 2809.
actions are included. Owing to e-ph coupling, polarons [10] Donhauser Z.J. et al.,Science 292 (2001) 2303.
[11] Kergueris C. et al., Phys.Rev.B 59 (1999) 12505.
can be formed inside the molecule. Polaron propagation
[12] Tans S.J., Verschueren A.R.M., and Dekker C., Nature
is the main mechanism of transport through perfectly
393 (1998) 49.
dimerised (semiconducting) molecules. The presence of
[13] Dash L.K.and Fisher. A.J.,J. Phys.: Conden.Matt. 13
amid-gapstateassociatedtoasolitondefectinthemid-
(2001) 5035.
dle of the molecule involvesdifferent mechanisms for the
[14] Peierls R.E., Quantum Theory of Solids (Clarendon
transport. For injection energies around the mid-gap Press, Oxford) 1955.
state, the delocalisation and deformation of the soliton [15] Heeger A.J. et al.,Rev.Mod. Phys. 60 (1988) 781.
isthe mainmechanismforelectrontransfer. Withlarger [16] Ness H., Shevlin S.A. and Fisher A.J., Phys. Rev. B 63
4
(2001)125422; NessH.andFisherA.J.,Phys.Rev.Lett. [19] Chao K.A. and WangY., J. Phys.C 18 (1985) L1127.
83 (1999) 452. [20] Bonˇca J. and Trugman S.A., Phys.Rev.Lett. 75 (1995)
[17] Samanta M.P. et al., Phys. Rev. B 53 (1996) R7626; N. 2566; 79 (1997) 4874.
Lang and P. Avouris, Phys. Rev. Lett. 81, 3515 (1998); [21] Sols F., Ann.Phys.214 (1992) 386.
Magoga M. and Joachim C., Phys. Rev. B 56 (1998) [22] Xie S. and Mei L., Phys. Rev. B 47 (1993) 14905; Sun
4722. X. et al.,Phys. Rev.B 35 (1987) 4102.
[18] Y. Lu, Solitons and Polarons in Conducting Polymers, [23] Bowler D.R. and Fisher A.J., Phys. Rev. B 63 (2001)
(World Scientific, Singapore) 1988. 035310.
Phonon modes
1
0.8 (a) (b) (c)
0.6
ment 00..24
Displace −−00..042
−0.6
−0.8
−11
0.8 (d) (e) (f)
0.6
ment 00..24
Displace −−00..042
−0.6
−0.8
−10 20 40 60 80 1000 20 40 60 80 1000 20 40 60 80 100
atomic site j atomic site j atomic site j
FIG. 1. Optic component of the phonon modes for a
chain length N = 99. The optic component is defined as:
(−1)j(V (j+1)−2V (j)+V (j−1)). (a) is themode asso-
λ λ λ
ciated with the soliton translation (¯hω =0.020 eV), (b) with
a
the deformation of the soliton width (¯hω =0.114 eV), (c) is
b
the so-called third mode (¯hω =0.134 eV). (d,e,f) are other
c
long-wavelengthopticmodes(¯hω =0.144,0.152,0.158eV).
d,e,f
0.15
0.10
0.05
n
o
ati
eriz 0.00
m
di
-0.05
-0.10
-0.15
0 20 40 60 80 100
atomic position j
FIG. 2. Dimerisation dj (in ˚A) for isolated neutral
molecules. Thin solid line: N = 100, for even N, dj is con-
stant (dc) in themiddle of thechain. Solid line: N =99, for
odd N, a soliton exists in the middle of the chain separating
two domains of dimerisation with opposite site. Dot-dashed
line: theabsolutevalue|dj|fromwhichdc andtheendeffects
are substracted (N =99). Thin dashed line: deformation in-
duced by adding an extra electron in thechain N =100; the
local reduction of the dimerisation representsa polaron.
5
E=0.00
81
osition’ i61 AB00O..00V45E84 -- 000...000566400
ectron ‘p41 000...000334062 --- 000...000344628
el 0.024 - 0.030
0.018 - 0.024
21 0.012 - 0.018
0.006 - 0.012
0.000 - 0.006
BELOW 0.000
1
20 40 60 80
atomic position j
Fig. 3(a)
E=0.41
81
on’ i61
ositi
p
n ‘
ectro41
el
21
1
20 40 60 80
atomic position j
Fig. 3(b)
FIG.3. Two-dimensional (atomic j/electron i positions)
map of thedimerisation d[i] (in ˚A) obtained from theatomic
j
distortions u[i] for the N =99 wire length. Here we plot the
j
absolute value |d[ji]| substracting dc and the end effects. The
incoming electron propagates from the left to the right. (a)
Injection at mid-gap (E =0.0): thebright feature represents
the soliton whose position and width vary for the different
electron positions i taken to calculate d[i]. (b) Injection at
j
E = 0.41 eV close the polaron resonance peak in the trans-
mission (Fig. 4): a virtual polaron is formed (bright feature
across the first diagonal). The maximum change in |d[i]| is
j
≈0.03−0.04˚Aaroundthepolaron and≈0.07˚Aaroundthe
soliton.
6
Electron transmission
0
−1
g T(E)10−2 −−21..04
o
l
−2.6
−3
−3.2
−3.8
0.00 0.05 0.10
−4
0 0.2 0.4 0.6 0.8
E [eV]
FIG. 4. Transmission probability T(E) versus the en-
ergy E for a molecular wire of length N = 99. Dashed line:
transmissionthrougharigidchaincontainingasolitondefect
(see the sharp peak at E = 0 corresponding to the soliton
state resonance). Solid line with circles: transmission when
the e-ph coupling is included. The delocalisation of the soli-
tonandthepolaron-solitoninteractionbroadentheresonance
peak at mid-gap into a “mini-band” (shown in the inset, en-
larged low-energy region). The polaron resonance peak ap-
pears around E ≈0.42 eV.
7