Counting Statistics in Nanoscale Junctions

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by  Yu-Shen Liu

Counting Statistics in Nanoscale Junctions.
Yu-Shen Liu and Yu-Chang Chen∗
Department of Electrophysics, National Chiao Tung University, 1001 Ta Hush Road, Hsinchu 30010, Taiwan
(Dated: January 14, 2011)
We present first-principles calculations for thethird moment of thecurrent in atomic-scale junc-
tions. We calculate this quantity in terms of the effective single-particle wave-functions obtained
self-consistentlywithinthestaticdensity-functionaltheory. Asanexample,weinvestigatetherela-
1
tionsamong theconductance,thesecond andthirdmomentsofthecurrentforcarbon-atom chains
1
of various lengths bridging two metal electrodes. We find that the conductance, the second-order
0
2 and the third-order Fano factors show odd-even oscillation with the number of carbon atoms with
thethird-orderFano factor positively correlated to the conductance.
n
a
J
Nanoscale electronics has generated a tremendous
3
wave of scientific interest in the past decade due to
1 (a)
prospectsofdevice-sizereductionofferedby atomic-level C C C C
] control of certain physical properties [1]. In addition,
l
l it has spurred great interest in the fundamental under-
a
standing of quantum transport [2]. One of these funda-
h
- mental questions relates to the moments of the current.
s For instance, the second moment - shot noise - defines 6
e
m the quantum fluctuations of the current at zero tem- (b)
4
perature due to the quantization of charge. Shot noise
.
mat crehaacrhgeesatnhdeIcliassstihcealalvimeriatge2ecIu,rrwehnetr[e3]e, wishtehneeelelecctrtoronns u.) 2
in a conductor drift in a completely uncorrelatedway as a. 0
d- described by a Poissoniandistribution of current events. X (
-2
n On the other hand, in the presence of a junction with
o
a narrow constriction, electrons, shot noise can be ex- -4
c
[ mpriessssioedn apsroSb2ab∝ilitPiesnoTfne(a1c−hTeing)enin-chtearnmnesloTfnth[4e].trIatnsis- -6-6 -4 -2 0 2 4 6
2
a powerful tool for the explorationof quantum statistics Z (a. u.)
v
of non-equilibrium electrons [5–12] and may provide a
7 FIG. 1: (color online) (a) Schematic of the four-carbon
0 means to explore also local temperature effects in nano-
atomicjunction. (b)Thespacialdistributionofpartialcharge
4 structures [11]. In fact, it has been recently employed to
densityforelectronswithenergiesneartheFermilevelsshows
3 characterize the signature of molecules/atomic wires in π-orbitalcharactersatVB =0.01V.Verticalblacklinescorre-
.
5 junctions [5, 6, 13]. spondtotheedgesofthejelliummodelandcirclescorrespond
0 Thehighermomentsofthecurrent(althoughmoredif- to atomic position.
0
ficult to measure and calculate) provide deeper insight
1
into the statistics of charge dynamics, and are there-
: capableofformingdiversestructuresincludingdiamond,
v fore more refined tools to characterize the signature of
i graphite, fullerenes, nanotubes and graphene. Recently,
X molecules in junctions. However, no studies have con-
experimentalists have shown the possibility to form car-
sidered higher moments of the current in truly atomic-
r bon atomic chains from graphite using a transmission
a scale systems. To address this issue, we have developed
electron microscope [14]. The carbon atom chains have
atheoreticalapproachthatcombinedwithstatic density
regularlypatterned electronic structures as a function of
functionaltheory(DFT),whichallowsustocomputecor-
the number of carbon atoms [15]. In this regard, they
relationsuptothethirdmomentofthecurrentinatomic
are among the few model systems in which theory and
junctions.
experiments can be reasonably compared.
We have then investigated the relation between the
Let us then consider a system at steady state subject
conductance,thesecondmoment(orshotnoise,S ),and
2 toabiasV =(E −E )/e,whereE andE are
B FR FL FR FL
thethirdmomentofthecurrent(wedenoteitwithS )for
3 the right and left electrochemical potential, respectively.
aprototypicalnanojunctionconsistingofanatomicchain
The system is described by the field operator
with different number of carbon atoms connecting two
metal electrodes as shown in Fig. 1(a). This is not just Ψˆ = aαEK(t)ΨαEK(r), (1)
an academic example since carbon is a versatile element αX,E,K
2
K incident from the left (L) and right (R) electrodes
are later computed self-consistently in the framework of
-12 -9 -6 -3 0 DFT [9, 16, 17].
*
C10=[C4] 1 (4) 2 (4) 3 (4) The current operator is defined as

1 2 3 Iˆ(z,t) = e~ dr⊥ dK dK
C9=[C4] 1 (4) 2 (4) 3 (2) mi Z Z 1Z 2
b. unit) C8=[C4] 1 (4) 2 (4) 1 2 3 ·ei(EEX11−EE22X)αtβ/~aαE†1K1aβE2K2I˜Eα1βK1,E2K2(r),(3)
r
s (a 1 2 where
e C7=[C4] 1 (4) 2 (2)
Stat I˜Eα1βK1,E2K2(r)= ΨαE1K1 ∗∇ΨβE2K2−∇ ΨαE1K1 ∗ΨβE2K2.
of 1 2 (cid:0) (cid:1) (cid:0) (cid:1)
y C6=[C4] 1 (4) At zero temperature, the average of current operator
sit gives the first moment,
n
De C5=[C4] 1 (2) 1
e~ EFR
<Iˆ>= miZ dEZ dr⊥Z dKI˜ERK,R,EK(r), (4)
EFL
C4 1 where the following expectation values have been used:

<aα† aβ >=δ δ(E −E )δ(K −K )fα, (5)
E1K1 E2K2 αβ 1 2 1 2 E
-12 -9 -6 -3 0
with fL(R) the Fermi-Dirac distribution function in the
Energy(eV) E
left (right) electrode.
We note that our wave-functions as obtained in the
FIG. 2: (color online) The density of states in the contin-
uum region for N-atom carbon chains sandwiched between framework of DFT calculations describe noninteracting
two metal electrodes at VB = 0.01 V. Zero of energy corre- electrons, where the effective single-particle wavefunc-
sponds to theleft Fermi level. tions haveboundaryconditions describingelectronsthat
are partially transmitted and partially reflected. Unlike
the formalism developed in, e.g., Ref. [4], which has in-
L(R) L(R)
where α = L or R; aEK (t) = exp(−iωt)aEK ; and out ordering, the current operator defined in Eq. (4) de-
aL(R) is the annihilation operators of electrons inci- scribes steady-states current where the time ordering is
EK
dent from the left (right) reservoir, satisfying the anti- notinvolved. Therefore,thecurrentcorrelationfunctions
commutation relations are defined in the time-unordered way [21],
{aα ,aβ† }=δ δ(E −E )δ(K −K ), (2)
E1K1 E2K2 αβ 1 2 1 2 S (ω)=2π~ d(t −t )eiω(t1−t2) <∆Iˆ(t )∆Iˆ(t )>,
2 1 2 1 2
Z
where β = L or R. The single-particle wave functions (6)
ΨL(R), describe electrons with energy E and momentum and
EK
S (ω,ω′)=(2π~)2 d(t −t ) d(t −t )eiω(t1−t3)eiω′(t2−t3) <∆Iˆ(t )∆Iˆ(t )∆Iˆ(t )>, (7)
3 1 3 2 3 1 2 3
Z Z
where △Iˆ(t) = Iˆ(t)− Iˆ ; S2(ω) and S3(ω,ω′) are the state current can be defined as S2 = S2(ω = 0) and
D E S =S (ω =0,ω′ =0),
two- and three-current spectral functions, respectively. 3 3
Thezero-frequency2ndand3rdmomentofthesteady-
e~ EFR
S (z ,z )=2π~( )2 dE dr dr dK dK I˜RL (r )I˜LR (r ), (8)
2 1 2 mi Z Z 1⊥Z 2⊥Z 1Z 2 EK1,EK2 1 EK2,EK1 2
EFL
3
and
e~ EFR
S (z ,z ,z )=(2π~)2( )3 dE dr dr dr dK dK dK
3 1 2 2 1⊥ 2⊥ 3⊥ 1 2 3
mi Z Z Z Z Z Z Z (9)
EFL
[I˜ERKL1EK2(r1)I˜ELKL2EK3(r2)I˜ELKR3EK1(r3)−I˜ERKL1EK3(r1)I˜ERKR2EK1(r2)I˜ELKR3EK2(r3)].
where the expectation values in Eqs. (6) and (7) have orem [18],
been calculatedusing the Wick-Bloch-De Dominicis the-
0, for n=odd,
<Aˆ Aˆ ···Aˆ >= (10)
n n−1 1 (cid:26) mn−=11ηn−m−1 <AˆnAˆm ><Aˆn−1···Aˆm+1Aˆm−1···Aˆ1 >, for n=even,
P
where Aˆ denotes either creation or annihilation opera- 1.4 a. u. inside the positive backgroundedge of the elec-
i
torsandη =−1(1)forFermions(Bosons). We notethat tronjelliumasshowninFig.1(b)[15]. Asitwasfoundin
alternativedefinitionsofFouriertransform,e.g.,theinte- Ref. [15] when the length of the wire is increased by one
gralswithrespectto(t −t )and(t −t )inEq.(6)and carbon atom, two electrons are added to the π∗-orbital
1 2 2 3
(7)mayleadtodifferentparametrizationoffrequenciesω as shown in Fig. 2. The odd-numbered chains have a
and ω′ in the spectral functions. However,in the case of higherconductanceduetohalf-filledπ∗-orbitalwhilethe
the steady-state current where ω = ω′ = 0, the zero- even-numberedchainshavealowerconductanceduetoa
frequency current correlations are independent of the full-filled π∗-orbital at the Fermi levels, as shown in the
choices of Fourier transform. We also note that Eq. (8) upper panel of Fig. 3. In the middle and lower panels
leads to the relations SRR = SLL = −SLR = −SRL, of Fig. 3 we show the influence of the number of car-
2 2 2 2
where SRR = S (z → ∞,z → ∞), SLL = S (z → bon atoms on the second-order F and the third-order
2 2 2 2 2
−∞,z → −∞), SLR = S (z → −∞,z → ∞), and FanofactorF ,respectively,inthelinearresponseregime
2 2 3
SRL = S (z → ∞,z → −∞), which are a consequence (V =0.01 V). We observe that F and F both display
2 2 B 2 3
of current conservation. Similarly, Eq. (9) leads to odd-even oscillation with the number of carbon atoms.
SR(R,L)R =SL(R,L)L =−SL(R,L)R =−SR(R,L)L, (11) In order to better understand the relation among the
3 3 3 3
momentsofcountingstatisticsinthecarbonatomchains,
For a single-channel tunnel junction with transmis- wehaveinvestigatedthedifferentialconductance(defined
sion probability T, the first, second, and third moment as G = ∂I/∂V), the differential second-order Fano fac-
of current are given by I ∝ T, S ∝ T(1 − T), and tor [defined as F = (1/2e)∂S /∂I], and the differential
2 2 2
S ∝ −2T2(1 − T), respectively. The result of the third-order Fano factor [defined as F =(1/2e)2∂S /∂I]
3 3 3
unordered third moment is consistent with the results fortheC4andC5wiresinthenon-linearresponseregime.
of time-unordered three-current correlations derived by We observethatthe differentialconductanceofC4chain
other groups [19, 20]. Finally, we define the second- increases as the applied bias increases, while the differ-
and third-order Fano factors (which is dimensionless) ential conductance of C5 chain decreases as the applied
in the small bias regime for steady-state currents as bias increases, as shown in the top panels of Figure 4(a)
F = S /(2eI) and F = S /[(2e)2I], respectively. As and (b), respectively. The increase (decrease) of dif-
2 2 3 3
a direct result, for the single-channel junction G ∝ T, ferential conductance with the biases for the C4 (C5)
F ∝(1−T), and F ∝−2T(1−T), respectively. chainisduetothehalf-filled(full-filled)π∗ orbitalatthe
2 3
As an example, we have investigated the counting Fermi level, where more (less) states are included in the
statistics in linear atomic chains formed by four to ten current-carryingenergywindowcreatedbyincreasingbi-
carbon atoms (denoted as C4 to C10) bridging between ases. The bottom panels of Fig. 4(a) and (b) show that
twometalelectrodesmodeledaselectronjellium(r =2). conductanceandF arestronglypositivelycorrelatedin-
s 3
Thedistancesbetweentwoneighboringcarbonatomsare dicating that the dominant eigen-channels for counting
2.5 a. u., and the end atoms of the chain are fixed at statistics have transmission probabilities T >0.5.
4
increases (decreases) as bias increases in even- (odd-)
numbered carbon atom chains. We observe that F and
) 3
2 /h 2.0 differentialconductancearesignificantlypositivelycorre-
2e 1.6 lated,thusshowingthatthird-orderFanofactorprovides
( 1.2
G more information than the second-order Fano factor re-
0.8
garding the transmission probabilities of eigen-channels.
The authors thank MOE ATU, NCHC, NationalCen-
0.2
F2 ter for Theoretical Sciences(South), and NSC (Taiwan)
forsupportunder GrantsNSC 97-2112-M-009-011-MY3,
0.0 098-2811-M-009-021, and 97-2120-M-009-005and M. Di
Ventra for useful discussions, and S. D. Yao for drawing
-03 .05
F Fig. 1 and 2.
-0.10
4 5 6 7 8 9 10
No. of Carbon Atoms
∗ Electronic address: [email protected]
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