Cross correlations in disordered, four-terminal graphene-ribbon conductor: Hanbury-Brown and Twiss exchange as a sign of non-universality of noise

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by  Zhenbing Tan

Cross correlations in disordered, four-terminal graphene-ribbon conductor:
Hanbury-Brown and Twiss exchange as a sign of non-universality of noise
Z. Tan,1 T. Nieminen,1 A. Puska,1 J. Sarkar,1 P. Lähteenmäki,1 F. Duerr,2
C. Gould,2 L. W. Molenkamp,2 K. E. Nagaev,3,4 and P. J. Hakonen1,∗
1Low Temperature Laboratory, Department of Applied Physics, Aalto University, Espoo, Finland
2Physikalisches Institut (EP3), University of Würzburg, Würzburg, Germany
3Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Science, Moscow, Russia
4Moscow Institute of Physics and Technology, Dolgoprudny, Russia
We have investigated current-current correlations in a cross-shaped conductor made of graphene
6
ribbons. We measured auto and cross correlations and compared them with the theoretical pre-
1
dictions for ideal diffusive conductors. Our data deviate from these predictions and agreement
0
can be obtained only by adding contributions from occupation-number noise in the central region
2
connecting the arms of the cross. Furthermore, we have determined Hanbury – Brown and Twiss

n (HBT) exchange correlations in this system. Contrary to expectations for a cross-shaped diffusive
a system, we find finite HBT exchange effects due to the occupation-number noise at the crossing.
J
The strength of these HBT exchange correlations is found to vary with gate voltage, and very a

1 distinct HBT effect with large fluctuations is observed near the Dirac point.
3

] Disordered graphene is an extraordinary tunable sys- resistance of the central region. The Fano factor, i.e.
l
l tem for studying electrical conduction ranging from the ratio between autocorrelation noise power S and
a nn
h nearly ballistic transport [1, 2] to hopping conductivity the Poissonian noise S = eI related to current I in
P n n
- [3–8]. In narrow graphene wires, in particular, the num- terminal n, is found to remain at F = 1/3, i.e. as for
s
e ber of transport channels can be varied significantly by a single wire, when biasing is done at terminal 1 and
m tuning charge density by gate voltage and conduction other terminals are grounded. In particular, the semi-
. can be pinched off fully near the charge neutrality point classical theory predicts additivity of cross correlations
t
a (CNP).Theelasticmeanfreepathcanbemaintainedrel- insuchacross-shapedconductor, whichwouldmeanthe
m ativelylargecomparedwithdevicedimensions,whilethe absence of Hanbury–Brown and Twiss (HBT) exchange
- importanceoflocalizationandCoulombinteractionscan effects [18] in our sample.
d
be varied by adjusting the charge density [9–11]. Disor- In this Letter, we report measurements on auto and
n
o deringraphenecanleadeithertoincreaseordecreaseof cross correlations which are compared with the theoreti-
c shotnoise[12–15]. Thus,ingraphenenanoribbon(GNR) cal predictions for ideal diffusive conductors [16, 18, 19].
[ systems,itispossibletostudyphysicsofcurrent-current Our data deviate from the noise predictions for diffusive

1 correlations in a regime where disorder can be tuned, systems and agreement can be obtained only by adding
v which makes it an excellent platform for investigating contributions from occupation-number noise in the cen-
0 noise properties of disordered conductors. tral region connecting the arms of the cross. The pres-
9
Shot noise originates from the granular nature of ence of this noise is in line with observed negative bend
2
charge carriers, and it can be used as an independent resistance, which indicates partly ballistic transmission
0
0 testfortheconductionmechanism[16,17]. Howeveritis across the centre of our graphene sample. Our exper-
. difficult to distinguish between different models of noise iments demonstrate the existence of HBT exchange ef-
2
in graphene using two-terminal measurements because fects with clear difference from the expectations for a
0
6 several of them give nearby strength, on the order of 0.3 regular diffusive system. The HBT effect varies substan-
1 - 0.4, when compared to Poissonian noise. One of the tially with gate voltage and it becomes very strong near
: ways to overcome this difficulty is measuring the cross- the CNP.
v
i correlated noise in multiterminal graphene systems. Inagraphenecrossdevice,theremaybedeviationsout
X
Theoretical treatment of current-current cross correla- fromtheframeworkofthesemiclassicaltheory. Thebasic
r
tionsS betweenterminalsnandminadiffusivecross assumptionofdiffusivetransporttheoryisthatthemean
a nm
geometryaswellastheirrelationtoautocorrelationshas free path (cid:96) (cid:28)min{L,W} compared with the dimen-
mfp
beenperformedinRefs. 18and19withvirtuallyequiva- sions of the sample, the latter condition of which is not
lentfindings. Inthesemiclassicaltheory[19],thespectral wellfulfilledinanarrowGNR.Deviationsfromfinitesize
density of noise in a diffusive system is governed by the effects are estimated in Ref. 18, which predicts a small
local distribution function. This function is sensitive to positive exchange term on the order of ((cid:96) /L)G eV
mfp 0
diffusion of electrons, which is dependent on the local for a metallic diffusive cross. This prediction, however,
conductance and geometry of the conductor. The semi- has the opposite sign with respect to our experimental
classical theory predicts similar behavior for shot noise results, which are more reminiscent of the behavior of
in all cross-shaped diffusive conductors with negligible a multiterminal chaotic quantum dot [20]. Our mea-
2
4
G
p
3 11
G
t
2
100 nm
FIG. 1. Scanning electron micrograph of the GNR sample;
light gray denotes the edges of the graphene. Terminals 1
and3wereemployedforcrosscorrelationwhilebiaswassup- FIG. 2. Conductance G=I/V vs. V at small V using bias
g b
plied via 2 and 4 in the HBT experiments. The white scale configurationC:IngoingcurrentsI andI arenegative,while
2 4
bar corresponds to 100 nm. The overlaid arrows define the I >0andI >0. TheinsetatV =−30Vdisplaysnegative
1 3 g
straight and bent carrier paths with conductances of Gp and bendvoltageV21,34,wherecurrentisfedbetweenterminals4
Gt, respectively. and 3 and the voltage is measured across terminals 1 and 2.
V Arm 1 Arm 2 Arm 3 Arm 4
g
surements, in fact, reveal non-local conductance which −10 V 22 20 22 24
−30 V 33 35 37 35
indicates that (cid:96) (cid:39) W and that the transport over
mfp
the central area of the cross for some charge carriers is
TABLE I. Arm conductances at gate voltages V = −10 V
ballistic. Furthermore, strong disorder scattering may g
and V =−30 V given in µS.
g
change the Fano-factor in graphene. Our results show
that the noise properties of the system can be accounted
for by the standard diffusive theory provided that the diffusive regime far away from the Dirac point display
central region is considered as a four-terminal ballistic symmetry within approximately ±6 % at V =−30 and
g
conductor with nonequilibrium electron distributions in ±9 % at V =−10. Some asymmetry in conductances is
g
theterminals,whichalsocontributestothenoisethrough observed at V ∼ 0 and V ∼ 15 V. However, the asym-
g g
the occupation-number fluctuations of incident electrons metry in these regions is bias dependent and its influ-
[21]. ence becomes reduced in the constant-current-level type
We investigated current-current cross correlations in of correlation determinations. The conductances Gp and
a disordered 530×530 nm2 cross-shaped graphene con- Gt, defining the behaviour in the central region, are es-
ductor. The length and the nominal width of the arms timated from non-local measurements and the geometric
amount to L ∼ 240 nm and W ∼ 50 nm, respectively. dimensions. The size of the central region, taken as a
A scanning electron micrograph of the actual measured square fitting within the middle of the cros√s, yields for
sample is displayed in Fig. 1. The ribbon samples were the relative direct conductance Gp/G0 =L/ 2W =3.4,
fabricated from micromechanically cleaved graphene on where G0 denotes the average arm conductance.
a heavily p-doped substrate with a 280-nm thick layer Even though the arms of our sample are undoubtedly
of SiO . Metallic leads to contact the graphene sheet diffusive,theballisticnatureofthecentralareashowsup
2
were first patterned using standard e-beam lithography in non-local conductance features, i.e. in negative bend
followed by a Ti(2 nm)/Au(35 nm) bilayer deposition. voltage [22–24]. The observed bend voltage illustrated
After lift-off in acetone, a second lithography step facili- in Fig. 2 is rather small but it indicates nevertheless
tated patterning of the GNRs. Our measurements down thatpartofthechargecarrierstraversethecentralregion
to 50 mK were performed on a dry dilution refrigerator. ballistically. The bend voltage Vbend can be calculated
using the Landauer-Buttiker theory. The result using
We first characterized the sample conductances. The
the parametrization of Fig. 1 reads
conductances of the arms were derived from the data for
I/V in Fig. 2 measured for the biasing configuration C, G (G −G )
where the biasing leads 2 and 4 are seen with negative Vbend = G (G +30G )+t 8Gp(G +G )Vb. (1)
0 p t t p t
(ingoing) current, while currents in 1 and 3 are positive
(see Fig. 1). Classically, setting the potential of the cen- The smallness of the measured bend voltage in Fig. 2
ter of the cross as V/2, we obtain the arm conductances (approximately a few per cent of bias voltage) indicates
given in Table I. The arm conductances in the properly that G =G within approximately ±20%.
p t
3
Ourcorrelationmeasurementsystemoperatesoverfre-
quencies f = 600−900 MHz [25]. This frequency is
BW
typicallywellaboveanyfluctuatornoiseduetoswitching
intransmissioneigenvaluesatthecontacts[26]. Still,the
employedfrequencyrangeislowenoughtocorrespondto
zero-frequency noise because the frequency is low com-
paredwiththeinternal1/RC scaleandthetemperature.
Analuminumtunneljunctionwasusedforcalibrationof
the noise spectrometers [27]. For IV curves with signif-
icant non-linearity, S values were first derived as a
nm
function of current in the limit I → 0. The resulting
reading of dS /dI was converted to dS /dV by us-
nm nm b
ing the measured differential conductance dI/dV . Note
b
thatwealwaystaketheoppositeofthecrosscorrelations
whenn(cid:54)=m, whichmakesS positiveasallthesenon-
nm
diagonal correlations are negative in a fermionic system. FIG.3. RatioofS /S vs. V withbiasappliedviaterminal
13 11 g
Our results on the auto and cross correlation power 1 having other terminals DC grounded. The two data sets,
show linear slope with current at small bias, which goes light and dark, relate to Vb ≶0, respectively: their difference
isindicativeofthesmalluncertaintyinthedata. Thedashed
weakly down at large bias where inelastic scattering
line indicates the result from our model with G /G = 3.4.
starts to take place. When inelastic processes are im- p 0
The inset displays the calculated behavior of S /S vs. the
portant(inelasticlengthl (cid:46)L),shotnoiseingraphene 13 11
in ratio G /G (G /G =1).
p 0 p t
is reduced by the most strongly coupled energy relax-
ation processes, i.e. either by impurity-assisted acous-
tic phonon collisions or by optical phonons [26, 28–30].
A deviation from the diffusive behavior is apparent
Inelastic processes were strongest in our work near the
in Fig. 3, which displays the measured ratio S /S
CNP, as rather large voltages were needed for biasing. 13 11
for our graphene cross. Our theoretical calculation for
Autocorrelation S was investigated with bias V in
11 b
G /G = 3.4 yields S /S = 0.275, which agrees well
terminal 1 and the other terminals grounded. In this p 0 13 11
with the experimental data away from the Dirac point.
configuration, we found F (cid:39) 0.4, close to the values re-
At V = −30...−10 V, the sample behaves as a uni-
ported in Ref. 31 for a configuration with floating side g
formsystem,wherecurrentisdividednearlyequallyfrom
terminals;similarlyF wasincreasedneartheCNPwhere
a single biased lead into the three other arms (see Ta-
the IV curves are strongly non-linear. This Fano factor
ble I). Consequently, the clear variation of S /S in
is higher than the universal value F = 1/3 for diffusive 13 11
this regime indicates non-constant value for G /G as a
systems. Away from the CNP, our value of F (cid:39) 0.4 is p 0
function of charge density, as otherwise S /S should
in agreement with the theoretical results for disordered 13 11
remain better as constant at |V | > 20 V. In the range
graphene ribbons [13]. In Ref. [31] it was concluded g
of V = 3...+13 V, in particular, the electrical trans-
thattheseresultsareinaccordancewithGaussiandisor- g
portisinfluencedbyhoppingconduction. NeartheCNP
der having a dimensionless strength of K ≈ 10, which
0
(8 V<V <10 V), we find a strong decrease in S /S
meant that the conductance is strongly affected by dis- g 13 11
which cannot be accounted for by our extended diffusive
order [13]. However, almost the same result is obtained
cross theory. Increased shot noise due to tunneling and
in our model with diffusive arms and ballistic central re-
localized states near the CNP is a likely contribution to
gion. Indeed, the calculations for G =G =3.4G give
p t 0
the strong decrease of S /S .
F =0.367forthefloatingsideterminalsandF =0.394 13 11
f g
for the three grounded ones [32]. Finally, we present our data on the Hanbury – Brown
For a symmetric diffusive cross with negligible resis- andTwisseffect. WedefinetheHBTexchangecorrection
tance of the central region S13/S11 = 1/3. Our calcu- terminaccordancewithRef. [18]by∆S =SC−SA−SB,
lated result for this ratio depends on the significance of where SA, SB, and SC denote the absolute values of the
the occupation number noise and the inset in Fig. 3 de- cross-correlated noise power spectra between terminals 1
picts the theoretical ratio as a function of G /G with and3inthreedifferentconfigurationsA, BandC:inthe
p 0
G = G . In the limit of G → ∞, our theory yields HBT configuration A, (B), bias was applied to terminal
p t p
the diffusive value S /S = 1/3, while in the limit of 2 (4) while the other terminals were connected to DC
13 11
G → 0 and for G = G , we obtain 2/9. Hence, we ground;inthecaseC,both2and4werebiasedand1and
p p t
expect a clear deviation from the diffusive value when 3 DC-grounded. In order to compare the experimental
the occupation-number noise of the central region plays results more accurately with theoretical predictions , we
a role, even though no asymmetry exists in the conduc- present the scaled HBT ratio ∆S/(SA+SB) in Fig. 4.
tion and G =G at the crossing. Our data display a strong modification of the HBT
p t
4
FIG. 5. Theoretically calculated HBT effect ∆S/(S +S )
A B
as a function of G and G ; the arm conductance is set to
p t
FIG. 4. HBT exchange correction ∆S vs. Vg obtained from G0 = 1. In our analysis we are using the overlaid trace for
low-bias cross correlation experiments extrapolated to Vb → ∆S/(SA+SB) on the diagonal at which Gp =Gt.
0. Thesolidlineindicatesourmodelresult∆S/(S +S )S =
A B
−0.175 using G /G = 3.4. The inset displays the linear
p 0
dependence of ∆S on Vb measured at Vg =−30 V. The strong growth of the absolute value of HBT
exchange effect near the CNP is presumably due to
Coulomb blockade and tunneling becoming more impor-
exchange factor near the Dirac point. The strength of
tant. For an island connected to four metallic leads by
this change, however, cannot be captured by our ex-
fourtunneljunctions,wehavemeasured∆S/(S +S )=
A B
tended diffusive conductor model. In our theoretical
−1 within 3%. We did also check whether enhanced
model (see Fig. 5), ∆S is seen to vary with the ratio
electron-electron interactions near the CNP could ac-
of G /G which is likely to be modified near the CNP
p 0 count for the enhanced HBT effect at small charge den-
point. Thereby, a moderate decrease in ∆S could be un-
sity. However, a calculation for the hot electron regime
derstood in terms of a stronger gate dependence in the
(without a ballistic center) yields ∆S/(S + S ) =
A B
conductanceinthecentralregioncomparedtothatinthe
−0.295, which is clearly different from the measured re-
arms. Such a mechanism, however, would only account
sults both near the CNP and far away from it.
for values |∆S|/(S + S ) < 0.30, which falls clearly
A B To conclude, we have studied cross correlations in a
short from our measured results.
diffusive, disordered graphene conductor where the elas-
Toaccountquantitativelyforourexperimentalresults,
ticmeanfreepathisontheorderofthefeaturesizeofthe
wehavecalculatedthenoiseinagraphenecrossusingthe
geometriclayout. Eventhoughthenon-localbehaviorin
semiclassicalBoltzmann-Langevinequationsfordiffusive
our graphene cross is quite weak, its presence pinpoints
arms,whichareconnectedviatheballisticcentralregion.
occupation-number noise that has a profound effect on
Due to the nonequilibrium distribution of electrons in
thecurrent-currentcorrelationsofthemultiterminalsys-
thearmsandtheoccupation-numbernoiseinthecentral
tem. As a consequence, the noise properties of this dis-
region, we obtain [32]
ordered system cannot be treated by standard diffusive
theories. By inclusion of the occupation-number noise,
20 3G +2G +10G
∆S =− G2G2 0 p t excellent agreement is obtained between our semiclassi-
3 0 t (G0+2Gp+2Gt)2 cal model and the measured noise properties, including
G G +2G G +2G2 the Hanbury–Brown and Twiss exchange effects in the
× 0 p p t t eV . (2)
(G +4G )4 b regime where charge density is large. Near the charge
0 t
neutrality point, Coulomb effects become strong and a
instead of ∆S = 0, that is given for regular diffusive large HBT effect with strong fluctuations is observed.
systembythesemiclassicaltheory[19], and∆S >0pre- We thank D. Golubev, C. Flindt, G. Lesovik, F. Li-
dicted for ballistic graphene [33]. The calculated result bisch, C. Padurariu, P. Virtanen, and T. Heikkilä for
for ∆S/(S +S ) is displayed in Fig. 5 on the plane fruitful discussions. This work has been supported in
A B
spanned by G and G . The regular diffusive behavior part by the EU 7th Framework Programme (Graphene
p t
∆S = 0 is obtained in the limit G → ∞. The dashed Flagship and Grant No. 228464 Microkelvin), by the
p
line in Fig. 4 is obtained from Eq. (2) using G = G Academy of Finland (projects no. 250280 LTQ CoE
p t
and G /G = 3.4. The agreement between the model and 132377) and Russian Foundation for Basic Re-
p 0
and the data is excellent in the regime where the charge search,GrantNo. 16-02-00583-a,theprogramofRussian
density in the sample is large. Academy of Sciences.
5
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6
Cross correlations in disordered, four-terminal graphene-ribbon conductor: Hanbury-
Brown and Twiss exchange as a sign of non-universality of noise – Supplementary
information
BEND VOLTAGE AND BEND RESISTANCE
Weadoptthefollowingmodeloftheconductingcrosswithaballisticcentralregion. Thefourarmsofthecrossare
assumedtobediffusivewithequalconductancesG , andthecentralregionisassumedtobeballistic. Moreprecisely,
0
there are 2m+n reflectionless channels originating from each arm, and n of them go straight ahead into the opposite
arm, while m channels turn left and another m turn right (see Fig. 1). Hence the central region may be treated as a
four-terminal conductor with the conductance matrix
 
2G +G −G −G −G
t p t p t
Gˆ = −Gt 2Gt+Gp −Gt −Gp  (S1)
 −Gp −Gt 2Gt+Gp −Gt 
−G −G −G 2G +G
t p t t p
where G = 2me2/h and G = 2ne2/h. If the electrical potential in arm i at the crossing is ϕc, the total current
p t i
flowing from this arm into the other arms equals
(cid:88)
I = G ϕc. (S2)
i ij j
j
On the other hand, this current is given by the Ohm’s law in the diffusive arm i
I =G (V −ϕc), (S3)
i 0 i i
where V is the external voltage applied to the outer end of the arm.
i
The two-terminal resistance between the opposite ends of the cross is calculated by setting V = V , V = 0, and
1 b 2
I =I =0. Solving Eqs. (S2) and (S3) for I readily gives
2 4 1
G +2(G +G )
R ≡V /I = 0 p t . (S4)
2t b 1 G (G +G )
0 p t
To calculate the bend voltage, it is sufficient to substitute V =V , V =0, and I =I =0 into the system (S2) -
1 b 4 2 3
(S3) and solve it for V and V . As a result, one obtains
2 3
G (G −G )
V =V −V = 0 t p V . (S5)
bend 2 3 G (G +3G )+8G (G +G ) b
0 p t t p t
The current flowing through arms 1 and 3 is
4G G (G +G )
I =−I = 0 t p t V , (S6)
1 3 G (G +3G )+8G (G +G ) b
0 p t t p t
and the bend resistance equals
1 G −G
R ≡(V −V )/I = t p . (S7)
b 2 3 1 4 G (G +G )
t p t
NOISE AND EXCHANGE EFFECT
The noise in the systems results from two sources. One of them is impurity scattering of nonequilibrium electrons
in the diffusive arms, and the other one is the noise in the central ballistic region. Though ballistic conductors
withequilibriumelectrodesdonotgeneratenoiseunderanappliedvoltageatzerotemperature,theyproduceitifthe
electrondistributionsintheelectrodesarenonequilibrium. Here,thediffusivearmsserveasnonequilibriumelectrodes
7
for the central four-terminal ballistic conductor, and therefore its contribution to the noise must be also taken into
account. The Langevin equations for the fluctuations of current in the diffusive arms are of the form
dδϕ
δI =δIext(x )+LG i, (S8)
i i i 0 dx
where δIext is the extraneous Langevin current, x is the coordinate along the arm, L is its length, and δϕ (x) is the
i i
fluctuation of electric potential in this arm.The correlation function of the Langevin currents is
(cid:90)
(cid:104)δIext(x)δIext(x(cid:48))(cid:105)=4δ δ(x−x(cid:48))LG dεf (x,E)[1−f (x,E)]. (S9)
i j ij 0 i i
An integration of Eq. (S8) over x with the condition δϕ (0)=0 brings it to the form
i
(cid:90) L dx
δI =G δϕc+ δIext(x), (S10)
1 0 i L i
0
where δϕc ≡ δϕ (L) is the potential fluctuation at the crossing. But on the other hand, the fluctuation of current
i i
flowing from arm i into the rest of arms equals
δI =(cid:88)G δϕc+δI˜ext, (S11)
i ij j i
j
where δI˜ext are extraneous random currents generated at the crossing due to the nonequilibrium distribution of
i
incident electrons with the correlation function
(cid:90)
(cid:104)δI˜extδI˜ext(cid:105)=2G dE[fc(1−fc)+fc(1−fc)], (S12)
i j ij i i j j
where fc(E) is the distribution function of electrons in arm i at the crossing. Its values may be obtained from a
i
system of equation similar to Eqs. (S2) and (S3) with fc in place of ϕc and f (E−eV ) in place of V , where f (E)
i i 0 i i 0
is the Fermi distribution function. The distribution function of electrons in the arms is governed by simple diffusion
at a given energy and presents a linear combination of distributions at its ends
(cid:16) x(cid:17) x
f (x,E)= 1− f (E−eV )+ fc(E). (S13)
i L 0 i L i
The system of equations (S10) and (S11) has to be solved for δI and δI , and then the correlation function S =
1 3 13
−(cid:104)δI δI (cid:105) has to be calculated using Eqs. (S9) and (S12).
1 3
First of all, we calculate the two-terminal Fano factor for the case where the current flows only through arms 1 and
3, whereas side arms 2 and 4 are floating. Hence one has to set V =V , V =0 and I =I =0 for the averages and
1 b 3 2 4
δV =δV =δI =δI =0 for the fluctuations. This gives us the Fano factor in the form
1 3 2 4
2 6G2+9G (G +G )+4(G +G )2
F ≡S /|eI |= (G +G ) 0 0 p t p t . (S14)
f 11 1 3 p t [G +2(G +G )]3
0 p t
It is easily seen that F depends only on the ratio between G +G and G . It tends to zero as for a purely ballistic
f p t 0
system when this ratio is small and approaches the 1/3 value for a diffusive conductor when it is large. As shown
√
in Fig. S1, it passes through a maximum F ≈ 0.48 at (G +G )/G = ( 5−1)/2 ≈ 0.62 and equals 0.367 for
f p t 0
G =G =3.4G . Thisislargerthantheshotnoisefordiffusivecontactswithpurelyelasticscattering,butsomewhat
p t 0
smaller than the noise in the hot-electron regime.
Inaconfigurationwherethevoltageisappliedtoterminal1andtherestofterminalsaregrounded,onesetsδV =0
i
for all i. The general expression for the Fano factor is too cumbersome, and we present it here only for the particular
case of G =G , where it reads
p t
F ≡S /|eI |= 2G 18G20+45G0Gp+32G2p. (S15)
g 11 1 3 p (G +4G )3
0 p
The F (G /G ) curve is similar in shape to F (G /G ) but lies higher and reaches its maximum F = 0.65 at
g p 0 f p 0 g
G /G ≈0.24. For the particular values G =G =3.4G , F equals 0.394.
p 0 p t 0 g
8
FIG.S1. DependenceoftheFanofactorontheratiobetweentheconductancesofthecrossingpointandthearmsforG =G .
p t
The solid line corresponds to F for the floating side terminals, and the dashed line corresponds to F for the three grounded
f g
terminals.
In a similar way, one calculates SA for V =V =V =0 and V =V , SB for V =V =V =0 and V =V , and
13 1 3 4 2 b 13 1 2 3 4 b
SC for V =V =0 and V =V =V . The resulting exchange term in the noise equals
13 1 3 2 4 b
20 G2G2(10G +2G +3G )(G G +2G G +2G2)
∆S ≡SC −SA −SB =− 0 t t p 0 0 p p t t eV . (S16)
13 13 13 3 (G +2G +2G )2(G+0+4G )4 b
0 p t t